End phase

[Maxpayne03]

my opponent got 8 cards in hand, he declared his discard phase and ask me if I got any instant to cast before his turn ends. Can I wait for him to discard first and cast an instant or use an activated ability afterwards?

[rarehuntertay]

From Comprehensive Rules:
514.3. Normally, no player receives priority during the cleanup step, so no spells can be cast and no abilities can be activated. However, this rule is subject to the following exception:
514.3a At this point, the game checks to see if any state-based actions would be performed and/or any triggered abilities are waiting to be put onto the stack (including those that trigger "at the beginning of the next cleanup step"). If so, those state-based actions are performed, then those triggered abilities are put on the stack, then the active player gets priority. Players may cast spells and activate abilities. Once the stack is empty and all players pass in succession, another cleanup step begins.

Since discarding down to 7 occurs in cleanup step, no you cannot activate spells/abilities unless there is a state-based action/trigger that can be responded to. Then you can.

[Maxpayne03]

However, he insists that if I do not cast any instant or activate any ability, his turn ends and I can no longer do anything.

[cltrn81]

Priority passes at the end of the second main phase.  Do your thing then.

[Maxpayne03]

The thing is I wanted to wait for his to discard his 8th card before casting my instant.

[particle]

The thing is I wanted to wait for his to discard his 8th card before casting my instant.

You can't wait until he moves to discard, but anytime he wants to change phases he needs for you to also want to change phases. He can't end his turn and move to discard without giving you chances to interact.

[InfinitiveDivinity]

The thing is I wanted to wait for his to discard his 8th card before casting my instant.

Why not just do it on his upkeep?