SQL Server Database Size Requirements and Design Considerations

First, you need CREATE DATABASE permissions to create a database. The db_creator role may be best for you in this case. Second, you need to script the restore.

For example:

RESTORE DATABASE someDatabase
FROM DISK = N'D:\sql\backups\myDatabase.bak'
WITH STATS = 10 ; 
GO

By selecting SQL user without login within SSMS' create user dialog, this is the T-SQL that is generated (with no other options besides the user name supplied):

USE [AdventureWorks2012]
GO
CREATE USER [User1] WITHOUT LOGIN
GO

If you were to create a login and attempt to alter the user with the new login, like the following attempt:

create login TestLogin1
with password = 'p@$$w0rd';
go

use AdventureWorks2012;
go

alter user User1
with login = TestLogin1;
go

You will get the following error:

Msg 33016, Level 16, State 1, Line 8
The user cannot be remapped to a login. Remapping can only be done for users that were mapped to Windows or SQL logins.

Because of that, I don't think you can link up a login to this type of user without recreating it. It should be relatively painless, as you could just script out the user, but change the parameters from WITHOUT LOGIN to whatever login you want it to be mapped to.

EDIT: As pointed out by Aaron below, ensure that you're taking note of permissions and role memberships before making any changes. You'll want to ensure that you can get back to the permissions state you currently are in, if that's the desired behavior.