Does the unique ID of an Armory wallet give any information about it?
According to the source code the wallet ID is an address byte and the last 5 bytes of the first address, encoded as base 58. The last 4 bytes of the address are part of a SHA-256 hash of the first part of the address. The address itself is the private key hashed once with SHA-256 and once with RIPEMD-160. So the only information contained in the Wallet ID is checksum information, which is also contained in your publicly available address. The address is available since it is the one you transferred your coins to.

I'm pretty certain that having the wallet ID visible isn't going to leak any information.