You are confusing an office with the duties it requires one to perform.
Truman holds an office which obliges him to perform certain duties as they arise.
Thus, Truman is referring to a particular duty/action which he must now perform, or that has become necessary for him to perform.
The M-W gives one of the main definitions of duty:
duty
obligatory tasks, conduct, service, or functions enjoined by order or usage according to rank, occupation, or profession (duties that he knew he would have to do— Joseph Conrad)
The phrase it becomes my duty is almost a stock phrase, or at least it was. Google-book the phrase and you will see dozens of instances of it. Most often with regard to an office or obligation that one has or holds.
As for become, it can mean to befit, be proper to. But this meaning does not fit. I suggest you check dictionaries to see that this definition almost always has a person as its object. This sweater becomes me. Or it refers to an action that is unbecoming to someone. But not unbecoming to something.