sql server express lan
The account which runs the SQL Server instance doesn't need admin rights on the computer and neither does the account which will manage the SQL Server instance.  If you plan on creating databases within the instance you'll need to make sure that you have sysadmin rights within the instance but this isn't related to the OS rights at all.  The person doing the install will need to make sure that they give you the rights within the database instance that you need before they leave.

The person doing the install that has admin rights will need to enable TCP/IP on the instance as that does require admin rights to do.