If you can login as root (note: on SSH this should be prevented; use
su or
sudo instead) of course you can change the password, with
run as root. I don't think SSH prevents that by default (then again, I see no reason why it could not be prevented).
Also I don't think you need to do anything else; the services running as root are usually setup by root to do that, so it should be enough to just change the password. I know on Windows you change a password for user, you need to go and re-change that for every service that needs password; it sucks.