Presuming you will be using TAR to "zip and copy", things like:
--keep-directory-symlink
-p, --preserve-permissions, --same-permissions
-P, --absolute-names
Whatever you do, you need to compare the files in VPS1 with the copied files in VPS2 (at least visually) before switching the IP in DNS.
[and maybe be prepared to switch the IP back if anything goes really wrong]