My opponent cast Murderous Compulsion, which says "Destroy target tapped creature". In response, I cast Aim High, which says "Untap target creature. It gets +2/+2 and gains reach until end of turn. (It can block creatures with flying.)"
Does my creature gets destroyed? I think not because of the stack, being my spell was the last to be cast.
That is correct, the target is essentially checked twice for validity, once while the spell is cast and again before it resolves. If the tapped creature is untapped before the spell has resolved then it is no longer a valid target and Murderous Compulsion will "Fizzle" (become countered), having no effect. Rule 608.2b covers this:
608.2b If the spell or ability specifies targets, it checks whether the targets are still legal. A target that’s no longer in the zone it was in when it was targeted is illegal. Other changes to the game state may cause a target to no longer be legal; for example, its characteristics may have changed or an effect may have changed the text of the spell. If the source of an ability has left the zone it was in, its last known information is used during this process. The spell or ability is countered if all its targets, for every instance of the word “target,” are now illegal. If the spell or ability is not countered, it will resolve normally. Illegal targets, if any, won’t be affected by parts of a resolving spell’s effect for which they’re illegal. Other parts of the effect for which those targets are not illegal may still affect them. If the spell or ability creates any continuous effects that affect game rules (see rule 613.10), those effects don’t apply to illegal targets. If part of the effect requires information about an illegal target, it fails to determine any such information. Any part of the effect that requires that information won’t happen.
Yes, you can save the creature in this situation by untapping it in response
Murderous Compulsion, like every other targetted spell or ability, checks whether its targets are legal both when you choose targets, as well as on resolution of the spell. If on resolution, the creature previously chosen as a target is now untapped, the target becomes illegal (and because there's only a single target involved, the spell gets countered by the game rules for not having any legal targets).
601.2c The player announces his or her choice of an appropriate player, object, or zone for each target the spell requires. (...)
608.2b If the spell or ability specifies targets, it checks whether the targets are still legal. (...) Other changes to the game state may cause a target to no longer be legal; for example, its characteristics may have changed (...). The spell or ability is countered if all its targets, for every instance of the word “target,” are now illegal. (...)
As for the stack - you would have to cast Aim High in response to your opponent's spell. Because the stack resolves from the top (last in, first out), your spell will resolve before theirs, untapping your creature before it would be destroyed.
405.2. The stack keeps track of the order that spells and/or abilities were added to it. Each time an object is put on the stack, it’s put on top of all objects already there.
405.5. When all players pass in succession, the top (last-added) spell or ability on the stack resolves. If the stack is empty when all players pass, the current step or phase ends and the next begins.
