Responsibilities during the Occultation

Our first and foremost duty towards our living Imam is to recognize him and become acquainted with him. We should constantly remind ourselves of his presence.
The Holy Prophet (saww) has said, "One who dies without recognizing the Imam of his time is like one who died during the pagan era (before the advent of Islam)". (Ilzaam an-Nasib fi ithbaat al-Hujjah al-Ghaib)
Imam Muhammad al-Baqir (as) has said, "One who dies without having (accepted) the Imam, it is as if he died in the pagan era, and people are not exempted from recognizing their Imam". (Muntakhab al-Aathaar)

How to constantly remind ourselves about the presence of our Imam?
Interestingly, we have been guided in this matter as well. Basically, we have been told to form certain habits which will make us question its philosophy and hence keep us aware of our Imam's presence. For instance:

To give Sadaqah (alms) for the safety of our Imam's life and constantly pray for his protection.
Not that the Imam (AJ) needs this. He is the proof of Allah (SWT) on this earth and will be protected by Him. But, it will be a way of showing our sincere concern for him and at the same time expressing our loyalty towards him.
The following dua' is highly recommended during our Imam's ghaybah:
O Allah, be for Your representative, the Hujjah, son of al-Hassan, Your blessings be upon him and his forefathers, a Guardian, a Protector, a Leader, a Helper, a Proof and an Eye until You make him live on the earth in obedience (to You) and cause for him to live in it for a long time.
Whenever the name of Imam (AJ) is mentioned, to stand up and place hand on the head saying three times: May Allah hasten in his re-appearance. This was the sunnah of Imam Ali Reza (as).
De'bal Khuza'i was a great poet of Ahlul-Bayt (as). He was once reciting a poem in presence of Imam Reza (as) when he read: "The re-appearance of the Imam is an undeniable fact. He will rise by the name of Allah with (all His) blessings.."
Imam (as) began to cry and said, "O'De'bal! Do you know this Imam?"He said, "No, O son of the Prophet! I have only heard that he will be from your progeny who will spread justice and equity on earth". Imam (as) said, "Listen! After me, my son Muhammad will be the Imam. He will be succeeded by his son Ali who will be succeeded by his son Hassan. And after him will be his son who will be the Mahdi awaited by the people". Having said this, the Imam (as) then stood up. Facing towards Kaaba, he placed his hand on his head and said thrice: "'Ajjalallahu Farajah" meaning "may Allah hasten in his re-appearance". (Nuqooshe Ismat)

Give importance to Fridays in hope of our Imam's re-appearance.
Friday is the day which Allah (SWT) has chosen for the re-appearance of Imam al-Mahdi (AJ). In the Ziyarah of Fridays, we recite:
"...and today is a Friday and it is a day when your re-appearance is expected.."
On this day, we should dress up well, wear perfume and make a point to attend the Friday prayer.

Recite the dua's related to or taught to us by our 12th Imam (AJ).
Such as Dua' Iftitaah in the holy month of Ramadhan, Dua' Nudbah which is to call the Imam (AJ), Dua' Allahummar Zuqna Tawfeeq at-Ta-'ah in which there are moral values to be learnt and inculcated etc...