Question: Trying to debug a stored procedure on a local SQL Server Express instance. I am running SSMS As Administrator. My login is in the sysadmin server role. My connection user is in the sysadmin server role. I get the message "Unable to start the Transact-SQL debugger, could not connect to the Database Engine instance 'localhost\sqlexpress'."

Answer: 
Yes - I switched to using a Windows Authentication connection and it works now. So the answer is Yes, debugging works with SQL Express.
I believe the Express version does not have the debugger. I personally have never found the debugger of value anyway because problems with t-sql tend to be of the "did I get correct results" variety (which are often debugged by trying different variation of where clauses and joins until you find the culprit). 

They are often do not concern problems of state while stepping through a series of steps. If you are doing something with many steps in one proc, then add a test or debug variable and write code to populate up what you want to see at that point in time (might be a variable, might be a select query, just depends on what you are trying to do) when running in test mode.