Acceptable answers are: As Soon As, Just After, Immediately After, Quickly, Early
Some example sentences are:
早々にあきらめる (to give up quickly)
早々とねる (to sleep early)
ホテルを出て早々 (just after leaving the hotel)
It seems to be mostly an adverb meaning quickly or immediately after something. It would be best for you to learn the accepted answers because they capture how you actually use the word. Soon or shortly or any moment now are used for future actions, this word isn’t. This is either an adverb meaning to quickly do an action, or a word meaning shortly after something happened something else happened. Shortly after would be better than shortly.
When I want to say soon, I will use もうすぐ or 間もなく as a starting place. They are words for things that will happen soon in the future.