Sunion RFID RFM-003 with Arduino

Sorry, I read more in that data sheet, and you may be up against a bit more than you signed up for...
Specifically, the data code looks to be self-clocked, with a fixed pulse width. You need to detect the pulse width yourself, and then clock the input according to that pulse width. The range is 250-320 microseconds per pulse, using some form of phase shift keying for 0 vs 1 data.
Interestingly they call this "Manchester code data," but last I learned, "Manchester" data is 1-high, 0-low. Also, given that the pulse width is in microseconds, I wonder if the reset/clock timings they specify in nanoseconds really should be in microseconds.

You probably want a 100 MHz oscilloscope with at least two channels, and/or a logic analyzer with similar performance, to figure this guy out.