How to chill a compote?
There is nothing especially interesting going on in a hot compote. You can chill it as quickly or slowly as you want to, until it reaches the desired temperature. Food safety requires that you don't keep it for more than 4 hours in the danger zone (between 60°C and 4°C), but if it is in individual serving bowls, on-the-counter chilling should be enough. Also, don't put large amounts of hot liquids in the fridge, you risk raising the entire fridge's temperature to unsafe levels, and the inside of a large container won't cool quickly enough to leave the danger zone. If you made lots of compote at once and want it very cold, chill it in an ice water bath. 

The final temperature can be whatever you prefer. I eat my compote slightly colder than room temperature (15°C to 20°C). You can eat it fridge-cold, room temperature, or even hot, if it tastes better to you.