No, 趁早 does not exactly mean ASAP (as soon as possible).
趁 means to take the opportunity
早 means early
So, 趁早 means take the opportunity while it is still early or at the earliest.
Just to clarify further:
趁早 carries an emphasis on the window of opportunity. There is a deadline when such opportunity will be lost. The urgency of the matter is conditional on the window of opportunity. That is, it can be a very urgent matter that needs to be handled immediately, OR it can be a non-urgent matter that you can take your time as long as the opportunity still exist.
If you think that this job is not
suitable for you and want to quit, let me know at the earliest.
Looks like it is going to rain again soon, better keep all the laundry that you have just hanged out to dry while it is still early.
A better translation for ASAP would be 尽快.
if someone asks me "你什么时候想去吃饭?
You can say "我想尽快去吃" to mean you would like to have your meal ASAP or "现在" to have your meal now if you are really in a rush.