I have a Freewheel cassette, 9-speed, 13/32T. Currently, I'm using a MicroNew 9-speed RD and a 9-speed shifter, but both of them aren't great, and I want to replace them.
All I can find is the SHIMANO Alivio M3100 9-Speed Long Cage Rear Derailleur (ERDM3100SGS RD) and SHIMANO Alivio M3100 9-speed shifter.
However, the Alivio M3100 9-speed RD specs state that the top sprocket max is 12T, while my current sprocket is 13T. I wonder if it's compatible or not
Normally you set the b-gap (gap between the guide pulley and the cogs) looking only at the lowest gear combination (small chainring, biggest cog). It's fairly likely that to make this rear derailleur work, you would need to screw in the b-tension screw beyond its ideal adjustment to gain the clearance needed. RD-M3100 has a non-offset guide pulley, and derailleurs like this are more sensitive to small cog clearance because one of the functions of offset is that when you shift off of the smallest chainring, the offset creates extra clearance (increases the b-gap), but with non-offset the gap is constant across all the rings. If you make the b-gap larger than necessary, it will probably still basically work, but there will be a shifting performance loss.
RD-R3000-GS is a current production part that would be a good choice here, and it will work with any Shimano 9-speed shifter. Its total capacity is 43t, and since your rear end is only taking up 19 of that in this case, it will be able to work with basically anything up front (total variance of up to 24 teeth between small and large rings).
