The pronunciation of ASAP depends on the person. Some people pronounce the individual letters A-S-A-P and others say ay-sap. Whichever you use, you'll generally be understood by a native speaker. Persoanlly, I use the former, so I would say, "Can I get this A-S-A-P?" (rather than ay-sap) mainly because I say "F-Y-I" and "M-Y-O-B" (Mind Your Own Business), but NASA is nah-sah. English is weird like that.
In the States we pronounce it エイサップ like "a" (the letter) + "sap". You sometimes hear people spell it out (エイエスエイピ) but that's less common and tends to add more emphasis making it more of a command than a request.
I agree with juern. Why are we explaining English abbreviations at a Japanese learning site? We should be explaining Japanese ones like ky, jk, kwsk, mjk, thpr, pgr...etc.
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