If you are arriving by train from Flagstaff, you will likely being coming into Union Station, not the airport. From there public transportation to your hotel is easy and cheap.

Go to metro.net and use the "Trip Planner" to plan your route. It looks like you'll take the Metro Red Line to the Vermont/Beverly station and the catch the #14 bus to the corner of Beverly Blvd & Genesse Ave. Then just walk one block to your hotel.

The airport is more tricky. I would spring for the Super Shuttle which quoted your hotel at $25 (supershuttle.com). But maybe someone who travels from that area to the airport regularly will have a better suggestion.

I agree that your hotel is in a good location to see the sights. Make sure to check out the Farmer's Market right near your hotel. Have fun!