You don't want or need putty for remote access to the db.
Do you have permissions to connect to the remote database from your local machine?
If so, you can just use
mysqldump --host=remotemachine --user=remotesqluser --password=remotepassword --databases databasename>database.dump on Vista, then load it onto your local box with
mysql --port=3307 --host=127.0.0.1 --user=localsqluser --password=localpassword <database.dump - if you were running linux locally I'd just pipe one program into the other, but I don't know if Vista's cmd.exe supports that.
If you can't connect remotely but have admin access to the linux db, you can give yourself permissions (at the remote mysql command line) with
grant select on databasename.* to 'remotesqluser'@'yourexternalIPaddress' identified by 'remotepassword'; (I think
select is all you need). Once you have access, you can copy the database as above. Use
revoke to lose the privileges when you've finished.
If you can't do any of this, you will need putty: Connect to the linux box and run the
mysqldump command there (with
--host=127.0.0.1). Copy the dump file to your Vista machine by whatever means you like - scp, ftp, http if you have a handy web server - and then you can load the database locally as above. Don't forget to delete the dump file on the linux box after you're done.
NOTE if your mysqldump options aren't set up sensibly, you'll probably want to add
--extended-insert to the command to speed things up (allows usage of multi-row inserts), and if there are any stored procedures in the database to transfer,
--routines as well.