NFS usermapping is a deep dark labyrinth leading you into NIS, NIS+ and LDAP. Do you want to explore this, or do you just want something that works?
You can certainly swap the usernumbers, as long as you also chown the files - I can give you a recipe for that if that's helpful.
Yes, that's pretty much it. The NFS server doesn't care about user names; it only sees the UID (and GID) of the process requesting the file access. The standard Unix rules about file permissions apply; if your (NFS server-side) file is UID 1000, GID 1000, rw-r----- then UID 1001 (no matter whether client or server) cannot write to the file.
[snip]You could do a number of things, but the simplest may be just to ensure that both your Raspberry PI UID 1001 and your server UID 1000 belong to the same GID, and make the appropriate GROUP permission changes on the server.
-- Lew Pitcher "In Skills, We Trust" PGP public key available upon request
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