Just add the user to the
sudo adduser <username> sudo
The change will take effect the next time the user logs in.
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):
# Allow members of group sudo to execute any command
%sudo ALL=(ALL:ALL) ALL
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 sudo lpadmin sambashare
Because your system configuration may vary, I suggest taking a look at the output of
groups <username> to see what groups are normally in use.