My guess is that the cast effect would go on the stack & resolve prior to Ulamog being on the field and in play as a legal target for Path, but I'd like some confirmation to be certain.
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You can exile your opponents' Plains before Ulamog resolves, but you can't stop your opponent from getting mana from them. Assuming no one has anything else to do besides the things you mentioned, it'd go like this:
From your question it sounds like you might be thinking of lands and mana as being the same thing, but as that sequence of events pretty clearly shows, the lands are separate from the mana they produce.
No, you cannot prevent your opponent from tapping their Plains for mana.
Specifically, the play you describe plays out in the following sequence of events: