If you're not in control of the server, just use the single connection. However, best to create a new user called say web_user. Give it SELECT, INSERT, UPDATE and maybe DELETE privileges only to that one database.
Set up your OWN table with user names and passwords for each, and set a session variable like $_SESSION['username']='fred'. If fred adds any records to any tables, make sure that you have a field in the record called "Creator", and be sure you add
SET..
Creator = '" . $_SESSION['username'] . "',
... etc.
so you can keep tract of whodunit. :-)
Also, mysql doesn't care how many people are on the site (nor does apache), only how many people are concurrently requesting a connection. Since scripts usually execute in a second or less, it's instantaneous load you're worried about. Don't think this will be the biggest issue.
Remember though that in shared hosting there will be other loads like that subversive website about killing crawfish competing for your bandwidth :-)
If you need any further assistance let me know.
Sam