I guess I would try to buy them off stubhub, but I'm sure that pretty soon the Rams will have single game tickets for sale soon. I would call the ticket number at the Rams and ask them, telling them about the fact that you're coming on an international flight just to see them; I'm sure they will try to accommodate you.
The Rams play downtown, but downtown St. Louis is kind of boring. There are a lot of cool places in that town, but not downtown. If you just want to stay the night and not see the city, I would probably stay at the Lumineer (I think that's the name of it). It's a hotel/casino literally across the street from the stadium. If you are planning on staying more than one night, I would at least try to rent a car or something and go to some of the cooler areas. I would explore Forest Park, go to the Missouri Botanical Gardens, aka Shaw's Gardens (which are considered among the finest botanical gardens in the world), see the Arch, go to the Loop/Central West End/Soulard/downtown Clayton, etc. But if just staying one day, I would got to the Lumineer. It's also near Laclede's Landing which has bars and restaurants.
That's the best advice I can give you.
There is plenty to do downtown, especially for a visitor.
THe Lumiere is a fine suggestion for a hotel. I'd also recommend the Renaissance.
Things to do downtown:
City Museum (not a typical museum, more like a playground for the entire family).
If you like to bowl there is the Flamingo Bowling Alley downtown, pretty neat place.
The Cardinals will be in town that weekend, maybe try to get some tickets for a game there, check out Ballpark Village while your at it.
Check out the Old Court House where the famous Dredd Scott trial was held...or the Old Cathedral if you are interested in religious venues.
The city garden is downtown, it isn't something awe inspiring by any means, but always a nice area to walk through.
Of course, the Arch has a nice museum to check out and might as well ride to the top.
If Bars are your thing I will tell you I prefer the Soulard area, technically not "Downtown" but you can walk there if needed, a very short drive/cab ride. Other areas are Washington Avenue right by the dome. The Landing like mentioned above is another area, though it is more or a tourist and/or bachelorette/bachelor party type area. The 360 Bar is supposed to be a good time, but I don't have any first hand experience. The CWE, Clayton, U-City loop are also other great suggestions.
Food: Downtown Mike Shannons and Charlie Gittos are some well known spots. Both good but not sure either would be my first choice. For more casual places I think Bailey's Range is a fun burger and shake place. Also, Rooster is a good spot downtown for breakfast.
Like others suggested...if you plan on venturing out of the downtown area let us know...I can give you some other suggestions.
