Translation:My son will be an engineer when he finishes university.
üniversite bitirmek is perfectly proper. it means finishing a university. üniversiteyi bitirmek means finishing the university and in this example since the university is a known, specific university (his university) we use accusative form.
In English you can sometimes omit 'the' but that doesn't mean that the object is not specific and therefore the acusative should not be used in the Turkish translation. I would say always stick with the accusative as long as the English translation doesn't have an indefinite article or plural ending.
yes, 'the university' should be accepted.
by the way I just realized I made a small mistake in my comment. Üniversite bitirmek would translate to 'finish a university' while Üniversiteyi bitirmek would be 'finish university' or 'finish the university'. I edited my comment.
In this particular case, I think most people would understand it as being the same, meaning-wise, and many people probably use them interchangeably in everyday speech.
That being said, I might break it down like:
Will be: statement of fact or certainty, or expressing
Going to be: maybe implies a prior plan
I think in a lot of uses of "[noun] olmak," it comes off as " be/become a/an [noun]" in translation.
Ex. baba olmak = "to become a father"
Ex. doctor olmak istiyorum = "I want to be a doctor"
'Bir' definitely can be used to express the indefinite, but I feel like it's omitted a lot too, without drastically changing the meaning.