Just to further clarify, I've always considered Union Station to be the closest stop to the Capitol building. There are some streets to cross and stuff, but 15 minutes is being generous. A taxi is a waste, pretty much. It's really just a few blocks away. By the time you walk to where you can get a cab, you'd be halfway there if you just walk in the direction of the station. BTW, no, do not take a cab to BWI if that's what you meant. Cost a mint and won't save you much if any time.

Expect the trains to be crowded at that time of day, but Union Station is the end of the line so at least you should be able to get a seat. Make sure you get on the right train for the Penn Line. (There are three MARC lines that leave from Union Station, and anytime you go in there in the afternoon there are likely to be several different MARC trains sitting there waiting to go.)

I think arriving 4:45 sounds awfully iffy, especially with needing to check a bag! There's a several-minute bus ride from BWI Rail Station to the terminal, and you may have to wait a few for the next bus. I'd go for the earlier train mentioned above. But if you're okay cutting it closer, the later one should still work.

You should be able to be at the Capitol until around 20 minutes or so before train departure and still have plenty of time to get to the station and buy a ticket. You can buy the MARC ticket from the same machines as sell the Amtrak tickets.