Can wrist strap be attached to running PC?
As long as you have the proper type of strap you will be safe.  ESD (dissipative) straps have a large resistance (~ 1 M\$\Omega\$) in series so that in the chance that the person gets shorted out that they do not end up being wired into mains voltages.  The concern here isn't for the ESD , any form of conduction will solve your ESD problems, the concern is for safety.  If your boss won't shell out for a mat, just make sure he hasn't cheaped out on the wrist strap too to save a few pennies.!