Just add the user to the "Administer the system" (admin) group:
sudo adduser <username> admin
This works because /etc/sudoers is pre-configured to grant permissions to all members of this group (You should not have to make any changes to this):
# Members of the admin group may gain root privileges
%admin ALL=(ALL) ALL
You should be able to see the changes immediately in the Users Settings window.
As long as you have access to a user that is in the same groups as your "original" user, you can delete the old one.
Realistically, there are also other groups your new user should be a member of. If you set the Account type of a user to Administrator in Users Settings, it will be placed in at least all of these groups:
adm dialout fax cdrom floppy tape dip video plugdev fuse lpadmin admin sambashare
Because your system configuration may vary, I suggest taking a look at Users Settings --> Advanced Settings to see what groups are normally in use.