As cbmeeks said, it's going to be nearly impossible with a microcontroller. A Raspberry Pi would be fast enough, but not with an operating system.
It's possible to run code directly on a Raspberry Pi without an OS, like a microcontroller. You don't even need to worry about cycle timing to emulate a NES rom; you can rely on the M2 signal to know when to read the value on the bus. You're going to run out of I/O pins -- the Raspberry Pi only has 26 -- but you can reuse pins with latches/multiplexers. If you only want to emulate NROM games, you could use two Pis -- one for the CPU address space and one for the PPU (and maybe the CIC chip too, since you'll have a couple of spare pins.)
Another option might be something from the BeagleBoard family, such as the BeagleBone Black. These normally run Linux, like the Raspberry Pi, but it seems like it's easier to use them without an OS. The BeagleBone Black doesn't have enough GPIOs even for just the CPU, but you can always multiplex, and you might be able to repurpose some of the other pins, especially if you're running without an OS.