No, just a large town.
There is a bus stop via Greyhound in Cowley, WY, but that's as close as you will get to Cody from Rapid City using Greyhound.
391 miles
Approximately 360 miles
Billings, Montana (population 104,934) is the closest major city with a population of over 100,000. Gateway communities for Yellowstone National Park include Livingston, MT; Gardiner, MT; Cooke City, MT; Jackson, WY; Cody, WY; and West Yellowstone, MT. Bozeman, Montana (population about 34,000), about 90 miles north of the park, also the largest airport in the region.
Rock Springs, WY
The address of the Cody Heritage Museum is: Po Box 701, Cody, WY 82414
yes,because Cody is right beside the national park
Douglas, WY
According to google maps it's about 378 miles (or just over 6 hours).
The driving distance from Cody, WY 82414, USA to Old Faithful, Yellowstone National Park, Yellowstone National Park, WY 82190, USA is 333.08mi / 536.04km
It is 491.40 miles according to MapQuest.
Cody, WY is 53 miles from the east entrance to Yellowstone National Park.