Nope.
To play Xbox games on a 360, you need a hard drive in the console. This is for two reasons
A) All original Xbox's came with a 10GB hard drive, so for Xbox games to work, there needs to be at least 10GB of storage.
B) The 360 hard drive has the BC emulation files on it.
The memory in the 4GB 360 is treated as a really limited memory card. You can't install games to it for example. Before I made my hard drive (I went and flashed my own HDD to work, I flat refuse to pay MS's price gouging), I had to install Skyrim onto a micro SD card.