ExpertFlyer.com is a pay service that now offers Award & Upgrade Availability Search. One can search a route three days at a time, any connection. The relevant fare classes for United are:

First/Business - Upgrade with Miles/SWU/CR1 (R)
First/Business - Upgrade with Miles/SWU/CR1 for Elites (RN)

It is very easy to skim these results, to find capacity. (Big surprise, there's not much.) One can also advance fairly quickly to the next three day window, although it's not one click.

In United Flight Search Results each flight includes a "Fare Class" link which shows whether R, RN are available. Needless to say, this is a tedious way to go, bordering on openly hostile. Whether I bother to requalify for 1K again comes down to whether I can find interesting R, RN capacity, so you'd think United would want to streamline these searches.