Is a person's promise to pray for someone considered a trust that must be fulfilled? And is it a condition that this supplication be made in the absence of the person for it to be answered?

It is permissible for a Muslim to ask his Muslim brother to supplicate for him. If someone is asked to supplicate and he says, "In shāʾ Allāh (God willing)," then it is commendable for him to do so as a recommendation, not an obligation. This is because fulfilling a promise is commendable unless it involves harm. The phrase "in shāʾ Allāh" is an exception that does not necessitate fulfillment if the matter were obligatory, so how much more so for commendable acts. Supplicating for someone who requested it in their absence is a supplication made in their absence, which is hoped to be answered.