I don't know if there are any cabins at Fishing Bridge. There are lots of them just down the road at the Lake Hotel.
There are some at Old Faithful Lodge area.
The Lake Hotel is only two miles from Fishing Bridge, so it's fairly central on the east side of the Park.
West Yellowstone has lots of motels and will probably be your best bet. As stated, it's right on the edge of the Park and 14 miles from the junction that will take you north and around the loop to Fishing Bridge, or south and around the loop past Old Faithful to Lake Hotel and Fishing Bridge.
There is a middle road between the north and south loops, so you can take two days to see the north loop and two days to see the south loop.
We like to plan on lunch or dinner at Old Faithful and watch the geyser go off before or after we eat...about 1 hour between eruptions.
I recommend a tour of Old Faithful Lodge and one or two of the visitor centers...Mammoth at the north end for sure.
Take binoculars. you will be glad you did.
Cell phone service is not so good, but you can get good service up at the top of Mt.Washburn.
I recommend also that you fly into Jackson, rent the car there, and drive through Teton National park on your way to Yellowstone. The Tetons are unforgettable!