Does Notion Thief replace draw effects triggered on your opponents' turns even during their upkeep?

Notion Thief has a replacement effect (using the word 'instead') that applies whenever an opponent would draw a card. Replacement effects are not limited by whose turn it is — they apply at any time the specified event would occur, per CR 614.1.

During an opponent's upkeep, many effects cause card draws: the normal 'draw a card' step doesn't happen in upkeep, but triggered abilities like those from Howling Mine, Phyrexian Arena, or similar enchantments often trigger at the beginning of upkeep. All of these draws are replaced by Notion Thief's effect — the opponent skips the draw and you draw instead.

Note that Notion Thief's effect is a replacement effect, not a triggered ability, so it doesn't use the stack and can't be responded to in the normal sense (CR 614.5). The substitution happens as the draw event is about to occur.

Example: Your opponent controls Phyrexian Arena. At the beginning of their upkeep, its triggered ability resolves and they would draw a card. Notion Thief replaces that draw — they lose 1 life (that part still happens normally), but you draw the card instead of them.

This applies equally during opponents' draw steps, their upkeeps, their main phases, or any other time — the timing of the turn phase is irrelevant to Notion Thief's replacement effect (CR 614.1a).