Can you cast an instant during your opponent's end step after they draw for the turn?

There is an important mix-up here: the draw step and the end step are two entirely different phases. The draw step occurs near the beginning of the turn (after the untap and upkeep steps), while the end step is the first part of the ending phase, which comes at the very end of the turn. By the time the end step arrives, the active player drew their card long ago (CR 504.1).

That said, you absolutely can cast instants during your opponent's end step. After the end step begins, both players receive priority in turn order (CR 116.3). Any time you have priority, you may cast instants or activate abilities with the proper timing (CR 116.1, 509.1b). The end step is actually a very popular time to cast instants precisely because it is the last moment before your own untap step, letting you maximize the use of a spell or ability.

Priority works as follows during any step or phase: the active player (your opponent, during their turn) receives priority first. If they pass priority and you then pass priority without anyone taking an action, the step ends. If either player takes an action, priority resets and the active player gets it again (CR 116.4).

Concrete example: Your opponent is in their end step. They pass priority. You cast Brainstorm (an instant) to draw three cards and put two back before your own turn begins. This is perfectly legal — you had priority during the end step and Brainstorm has instant speed.

So to summarize: yes, you can cast instants in the end step, but that step has nothing to do with drawing for the turn — those are separate, earlier events in the turn structure.