The affected player won't be able to cast any noncreature spells until after the time that your "beginning of upkeep" triggered abilities are put onto the stack on your next turn.

The affected player may still activate abilities, play lands, and cast creature spells.

The target player (and any other player) may cast spells in response to the activated ability of Hope of Ghirapur. The ability won't affect those spells and it won't affect spells that the target player cast before you activated it. (In other words, the ability can't be used to counter a spell.)

If you skip your turn, the affected player will be unable to cast noncreature spells until you actually start a turn.

When a player leaves a multiplayer game, any continuous effects with durations that last until that player's next turn will last until that turn would have begun. They neither expire immediately nor last indefinitely.