Does 一屁股坐下来 just mean 'sit down' or is there some other meaning in there?
The literal meaning of 一屁股坐下来 is "sit down with one move of one's butt". It is a description of a manner which implies "abruptly with no hesitation"

There are similar adverbial phrases:

一手 (with one move of one's hand)

一手搶過我手中的香煙 (with one move of his hand, he grabs the cigarette off my hand = he takes the cigarette off my hand with one move of his hand)

一腳 (with one move of one's foot)

一腳把球射入龍門 (with one strike of his foot, he kicked the ball into the goal = he kicks the ball into the goal with a single strike)

"一屁股坐下来" would mean "he sits down with one move of his butt ", therefore it can be described simply as "sit down abruptly (with no hesitation)" 

"One move" implies "one(quick/sudden)move"