Suppose I have a Rhonas's Monument and a 1/1 creature token on the field and I play an Ornery Kudu, putting the -1/-1 counter on the token and giving it +2/+2 via the Monument trigger. Does the token die immediately, or can I choose to apply the monument first so that the token lasts until end of turn (as a 2/2)?
You have no choice: you must apply the monument ability first (and so the token will survive). This is because the monument ability triggers when you cast Ornery Kudu, and will resolve before Kudu can actually enter the battlefield at all to trigger its own ability.
If you do have two triggered abilities that trigger at the same time however, you do get to choose the order they'll resolve.
To add to Benjamin's answer, if multiple abilities would trigger at the same time but they are controlled by different players, the triggers are put on the stack in AP/NAP order (CR 603.3.). That is, the active player puts his triggers on the stack in any order, then the next player in turn order does so, etc. Thus the last player's triggers will resolve first, and the active player's triggers will be last.