Programming with Bits and Bytes
Pyth, 36 35 bytes

u%@[t_G/G2yGi_jGJ16JG)x"!&gt;&lt;@"H256z0


Test harness

The internal representation of the accumulator is an integer. This integer is mod-ed by 256 on each iteration, as desired. The operations performed are -G-1, G/2, G*2 and G converted to base 16, reversed, and converted back to base 10, where G is the accumulator.

I missed the line about ignoring everything else. This has been remedied. Thanks, @Dennis.