No. You would have to go to your agent to do this as UA ticketing would not necessarily know how much you paid.

Even though they print a 'full price' on the ticket, this isn't was has been paid. This may be done to facillitate flight changes when the need occurs (such as an outlying station in the middle of Africa). It is only when the ticktet gets back to the bean counters do they know what was paid.

If you want an H class ticket, my guess is you will need to buy it yourself. I presume your company has the consolidated rate and you want to buy up?