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During an upkeep, can you cast instant spells? Do you get priority? Or does that only happen when an effect triggers during an upkeep? 

Yes, you can. It happens unconditionally.

Players always receive priority during the upkeep step. You may cast instants and activate abilities during that time.

What about during the untap, draw, clean up, and combat damage steps?

Players always receive priority during every step except untap and cleanup.

In some cases it's even possible to get priority during cleanup.

Players never have priority during the untap step. 

The only step no player gets priority in is the untap step. Most player don't get priority during the cleanup step (the only time player do is if something triggers during the cleanup step or if the check for SBAs catches something during that step). Players get priority during the draw step after the active player has drawn for the turn, and players get priority during the combat damage step after combat damage has been dealt.