Below are random selections from the various Ziyarat's and Duas of Imam Al-Qaim (atfs). 

Hadith 1:

From Ziyarat Aal-e-Yaseen as taught by Imam Mahdi (ع) in a letter:

السلام عليك يا حجة الله و دليل إرادته

"Salutations upon You, O the Proof of Allah (swt) and evidence of his intentions"

يا مولاي شقي من خالفك و سعد من أطاعك

"O' My Master, wretched (unhappy) is the person who opposes You, and Happy is the person who obeys You."

[Source: al-Ehtejaj, Vol. 2, Pg. 318]

Hadith 2:

From the Dua taught by Imam Mahdi (ع) that reached us through the first Safeer of the Imam: 

فَصَبِّرْنِي عَلى ذلِكَ حَتّى لا أُحِبَّ تَعْجِيلَ ما أَخَّرْتَ وَلا تَأْخِيرَ ما عَجَّلْتَ وَلا كَشْفَ ما سَتَرْتَ وَلا البَحْثَ عَمّا كَتَمْتَ وَلا أُنازِعَكَ فِي تَدْبِيرِكَ وَلا أَقُولَ: لِمَ وَكَيْفَ وَما بالُ وَلِيِّ الأمْرِ لا يَظْهَرُ وَقَدْ امْتَلأتِ الأرْضُ مِنَ الجَوْرِ، وَأُفَوِّضُ أُمُورِي كُلَّها إِلَيْكَ 

"So grant me patience to endure that (i.e. Gaibat) so that I will not wish for hurrying -up that which you have delayed or delaying that which you have made early.

And I will not disclose that which you have concealed, and I will not search for that which you have kept secret and I will not dispute with you about your plan (Tadbeer) and I will not ask 'Why', 'How' and why is the Master of the Affairs (Wali-e-Amr) not reappearing while the earth is already filled with injustice/oppression. But I entrust all my affairs to You"

وَلَيِّنْ قَلْبِي لِوَلِيِّ أَمْرِكَ

"And soften my heart for the Master of Your Affairs" 

وَأَرِنا نُورَهُ سَرْمَداً لا لَيْلَ فِيهِ

"And show us His light for eternity which has no night/darkness in it" 

وَاجْمَعْ بِهِ الأَهْواءَ المُخْتَلِفَةَ عَلى الحَقِّ

"And unite through Him the different desires/whims upon the Truth" 

اللهم ولا تسلبنا اليقين لطول الامد في غيبته وانقطاع خبره عنا ، ولا تنسنا ذكره وانتظاره والايمان وقوة اليقين في ظهوره والدعاء له والصلاة عليه حتى لا يقنطنا طول غيبته من ظهوره وقيامه ، ويكون يقيننا في ذلك كيقيننا في قيام رسولك صلواتك عليه وآله ، وما جاء به من وحيك وتنزيلك ، وقو قلوبنا على الايمان به حتى تسلك بنا على يده منهاج الهدى والحجة العظمى ، والطريقة الوسطى ، وقونا على طاعته ، وثبتنا على متابعته واجعلنا في حزبه وأعوانه وأنصاره ، والراضين بفعله ولا تسلبنا ذلك في حياتنا ولا عند وفاتنا حتى تتوفانا ونحن على ذلك غير شاكين ولا ناكثين ولا مرتابين ولا مكذبين

"O Allah, do not take away from us the certainty (yaqeen) because of His long-lasting Ghaibah (occultation) and disconnection of His news from us.

And do not make us forget -

Remembering/mentioning Him,

And to wait for Him,

And to believe and have strong faith in His re-appearance (Zahoor),

And to pray for Him,

And to send blessings/salutations on Him,

So that his long-lasting occultation (ghaibah) will never make us despair of His reappearance and His rising."

And our faith in this will be just like our faith in the advent of Your Messenger ( صلواتك عليه وآله), 

And (just like our faith in) all that which has been conveyed by Him to us through Your Revelation and Divine communication. 

O Allah, strengthen our hearts upon believing in Him so that You will lead us, at His hands, to the path of true guidance, the greatest destination, and the moderate way. 

Grant us strength to obey Him, make us firm in supporting Him, 

And include us with His party, backers, allies, and those satisfied with his actions. 

Do not deprive us of this during our lifetimes nor at the hour of our death, but (please) make us die while we are upon this (faith), without doubt, and without breach of our covenant to him, and without suspect, or belying him."

اَبرِز يا رَبِّ مُشاهَدَتَهُ ، وَثَبِّت قَواعِدَهُ ، وَاجعَلنا مِمَّن تُقِرُّ عَينَهُ بِرُؤيَتِه ، وَاَقِمنا بِخِدمَتِه ، وَتَوَفَّنا عَلى مِلَّتِه ، وَاحشُرنا في زُمرَتِه

"O Lord, make His seeing come into view, and establish His foundation, and make us from amongst those whose eyes are delighted by looking at Him, keep us in His service (khidmah), and give us death upon His religion, and gather us (on the day of judgement) in His group."

The complete dua can be seen here: http://www.duas.org/allahummaarrifni.htm 

[Source: Kamal al-Deen - Shaikh Sadooq, Vol.1, Pg. 530 & Misbah al-Mutahajjid, Shaikh Tusi, Pg. 294]

Hadith 3: 

What to do during the period of Ghaybah?

قال زرارة: فقلت: جعلت فداك فإن أدركت ذلك الزمان فأي شئ أعمل قال: يا زرارة إن أدركت ذلك الزمان فأدم هذا الدعاء

In a narration from Imam Sadiq (as), Zurarah asked the Imam (as), “May I be sacrificed for You, if I reach the period of Ghaybah what act should I do?”

Imam Sadiq (as) said, “If you reach that period, then recite the following dua regulary:

اللَّهُمَّ عَرِّفْنِی نَفْسَك فَإِنَّك إِنْ لَمْ تُعَرِّفْنِی نَفْسَك لَمْ أَعْرِفْ نَبِیك اللَّهُم عَرِّفْنِی رَسُولَك فَإِنَّك إِنْ لَمْ تُعَرِّفْنِی رَسُولَك لَمْ أَعْرِفْ حُجَّتَك اللَّهُمَّ عَرِّفْنِی حُجَّتَك فَإِنَّك إِنْ لَمْ تُعَرِّفْنِی حُجَّتَك ضَلَلْتُ عَنْ دِینِی

"O Allah, make me recognize You, For if You do not make me recognize You, I will not recognize Your Prophet.

O Allah, Make me recognize Your Messenger, For if You do not make me recognize Your Messenger, I will not recognize Your Proof (Hujjah)

O Allah, make me recognize Your Proof (Hujjah), For if You do not make me recognize Your Proof, I will deviate from my religion"

[Source: Kamal al-Deen, Shaikh Sadooq, Ch. 33, Hadith. 24]

Hadith 4:

From a Salawaat taught by Imam Mahdi (عج):

اللهُمَّ أعطِهِ في نَفسِهِ وَذُرّيَّتِهِ وَشيعَتِهِ، وَرَعيَّتِهِ وَخاصَّتِهِ وَعامَّتِهِ، وَعَدوِّهِ وَجَميعَ أهلِ الدُّنيا ما تُقِرُّ بِهِ عَينَهُ، وَتَسُرُّ بِهِ نَفسَهُ، وَبَلِّغهُ أفضَلَ ما أمَّلَهُ في الدُّنيا وَالآخرةِ، إنَّكَ عَلى كُلِّ شيءٍ قَديرٌ

"O Allah, grant Him (i.e. Imam Mahdi عج ), concerning himself, his offspring, his followers (Shia), his subjects, his special ones, his common ones, his enemies, and all the inhabitants of this world, [give him] that which delights His eye and makes Him happy.

And make Him attain the best of that which He desires in this world and the Next Word.

Verily, You have power over all things."

Read it completely here: https://www.duas.org/imamezaman/zarrab_isfahani.htm

[Source: Misbah al-Mutahajjid, Shaikh Toosi, Pg. 406]

Hadith 5:

From the Dua taught by Imam Sadiq (ع) famously known as Dua al-Ahd which is to be recited in remembering Imam al-Mahdi (ع):

اللَّهُمَّ اجْعَلْنِي مِنْ أَنْصارِهِ وَأَعْوانِهِ وَالذَّابِّينَ عَنْهُ والْمُسارِعِينَ إِلَيْهِ فِي قَضاءِ حَوَائِجِهِ (وَالْمُمْتَثِلِينَ لِأَوامِرِهِ)* وَالْمُحامِينَ عَنْهُ وَالسَّابِقِينَ إِلى‌ إِرادَتِهِ وَالْمُسْتَشْهَدِينَ بَيْنَ يَدَيْهِ 

"O Allah, make me one of His amongst supporters, His helpers, those who defend Him, those who rush to Him to fulfill His needs, [those who submit to (follow) His orders]*, those to stand up to defend Him, those who get ahead of others (take precedence) in going towards His desire and those who are Martyred in front of Him."

*As found in other books such as Mafateeh al-Jinaan

[Source: Bihar al-Anwar, Vol. 99, Pg. 111]

Hadith 6:

From Dua al-Iftetaah which is recommended to be recited every night in the month of Ramadan:

We say the following in asking Allah through Imam al-Qaim (عليه السلام):

ويسر به عسرنا، وبيض به وجوهنا وفك به أسرنا، وأنجح به طلبتنا، وأنجز به مواعيدنا، واستجب به دعوتنا وأعطنا به فوق رغبتنا

"O Allah, through Him make easy our hardships, 

And through Him whiten our faces, 

And through Him release us from captivity, 

And through Him make our requests successful, 

And through Him carry out our commitments, 

And through Him answer our supplications,

And through Him give us above our desires."

[Source: Tahzeeb al-Ahkam, Vol. 3, Pg. 111]