Should I push myself against my own will to achieve things I don't even really care about?

No. Unless, of course your 'caring' is ambivalent and you wish to overwrite your will in favour of one kind of 'caring'.

Bear in mind, of course, that many things you may push yourself to against your natural inclinations are actually goals that benefit you directly (or via the status granted for dominant 'altruistic' acts). Sometimes the reasoning 'I will be penalised by society or the universe in general if I do not do it' is itself a good reason to care. Like you get to continue to eat if you do it.