I forget if you have to switch United.com to advanced mode anymore or not. That used to be required to see R. (Google search if so). From then all you have to do is look at your route on United.com (revenue ticket, not award) while you are logged in, then click the "details" dropdown and you'll see everything listed there. If R is 0 then there is no confirmed space, anything higher than 0, then you can book it then and there and confirm an upgrade as soon as you get a ticket number.
One point to note, if you're looking at an itinerary with multiple segments, then look at your longest segment first by itself. United.com will only show the lowest of all segments on the itinerary, so for example if you have DEN-LAX-OGG, if LAX-OGG has R open but DEN-LAX doesn't, then you'd still see R0 even though you'd clear instantly on LAX-OGG if you bought the ticket and went to the upgrade screen in your reservation.