I own an xbox 360 slim and I own Fable: The Lost Chapters and I play on an Xbox 360 Slim. It works great!.. Sometimes as with all regular Xbox games playing on Xbox 360 there is some slight lag, and some slight errors with audio, but nothing too major, and none too problematic to not buy this game.
It depends on what model of Slim you purchased. If you got the system with the 250GB HDD then all backwards compatible X-Box games will work on your system, including Fable:TLC. But if you got the system with the 4GB of internal memory then none of the old X-Box games will work.
...without adding a 250GB HDD. This is only because the original Xbox games are programmed with the expectation that the console has an internal hard drive. It's the PS3 "slims" that aren't backward compatible anymore, period. (Not a fan-boy either way... these are just facts)