Edit: There is now an in game clock feature. It can be enabled by going to options, gameplay, and then enabling the in-game clock. The below is old information and no longer relevant.
There is currently no way to view the time elapsed in game. The lack of this has led to numerous threads requesting the feature on blizzard's forums. There appear to be no plans currently to add it, but I would not rule out the possibility of it being added eventually given all the requests blizzard is receiving.
I noticed earlier today that as of 1.1, the Blizzard maps were updated from v0.1 to v1.0. (Agria Valley is currently at v1.2.) Also, in the SC2 Editor, some Blizzard maps are showing up as not downloaded:
I haven't played 3v3 since pre-1.1, so all the updated 3v3 maps are still marked as Not Downloaded.
This supports Mag Roader's theory that the downloads consist of map data.
So, here's what I think it is:
When you join the matchmaking queue, the game asks Battle.net for a list of maps you might about to be playing on. If you have the latest versions of all the maps, this will probably only take a second.
If any maps are out of date, however, the game will update them. This will take a few seconds for each map since they're around a megabyte each.
This system would let Blizzard publish updated versions of maps, or add new ones to matchmaking, between actual patches.
Best Answer
Under your profile in the game(as well as outside the game at the Starcraft 2 website when you login to your account), there is a section called Match History where you can see all your previously played matches. This includes games from previous seasons. You can filter on 1v1, 2v2, 3v3, 4v4, FFA, Cooperative, and Custom game matches.
Unfortunately, this does not show you the name of your opponent until you click into the game in the Match History selection screen, so you'll have to remember the approximate time, game type and map on which you played the game to find previous matches.
Once you find the game, you can click through to find your opponent's profile and their rank. Unfortunately, you can only access the match score screen from within the game, not on the website, so you will need to boot up Starcraft 2 beforehand.