Is there any real difference between "prezent," "upominek," "podarunek," and "podarek?" Can they all be used interchangeably?
yes pretty much. I think prezent is most common, and upominek "feels" smaller, while "podarunek" and "podarek" feel old-fashioned
Thank you for your response!
It'd better say "World's Best Brother".
Present in Afrikaans
Does kubek actually ever mean cup (rather than mug)?
Well, according to https://en.wiktionary.org/wiki/kubek the word "kubek" apparently means a "mug (large cup)". In practice, a mug is a more likely gift than a cup, as it is easier to get mugs with some amusing sentiment applied.
This sentence is possibly past tense in meaning; can "to" ever be translated as "was"?
So, if I wanted a past tense Polish version of this sentence, I could only use "był prezentem"?
Yes, that's my understanding.