It IS for a specific area. Shinjuku is a city ward and every city ward have many areas/zones (think of them as neighborhoods) inside that all use their own postal code.
For instance:
Tokyo → Shinjuku → Nishi Shinjuku (West)
Tokyo → Shinjuku → Minami Shinjuku (South)
Tokyo → Shinjuku → <Area name here>
Etc
Every area name also have block numbers inside (called chome in Japanese)
For instance:
Tokyo → Shinjuku → Nishi Shinjuku 1-Chome (一丁目
Tokyo → Shinjuku → Nishi Shinjuku 2-Chome (二丁目
These are another part of the address though, usually 3 numbers, like 3-7-8 (3 = 3-Chome).
You can read more about it here (helpful if you gotta send stuff):
http://www.japan-guide.com/e/e2224.html
http://tokyo.angloinfo.com/information/housing/post-codes/
I don't know/think it's normal to use a generic ward postal code (like 160-0000) but I've used the wrong postal code before and still gotten my stuff. Even used a Shinjuku postal code when I live in Minato (160 vs 106). But 160-0000 is to identify a range in an area, and points to different places in Shinjuku (probably random) so I wouldn't use it if I were you.
If you're asking because you don't know the specific postal code, I would say it's okay to use the wrong one, but if you already have parts of the address, it should be easy to figure out at least a near-by postal code you could use.
Did you make an order mistake somewhere since you're asking? I can't guarantee it but it *should* be fine, as long as the rest of the info is correct. Dunno if they sort mail by postal code though, so maybe it'll take slightly longer.