The question you should be asking is: does a 9 speed cassette work with your shifter?
I think it's quite rare to find non-freehub rear hubs for 9 speed systems. Thus, I suspect you may have a 7 speed (or less) system. These systems have shifters with less positions than 9, and besides the cassette sprocket spacing is different, so no, your shifter most likely does not work with a 9 speed cassette.
If you have a 9 speed shifter (would be quite unusual for non-freehub systems), then yes, it will work (if Shimano).
The cable pull ratio on all 9-speed-or-less Shimano systems is the same (with the exception of some old Dura-Ace models) so your derailleur will work with a 9-speed shifter, and with an 8-speed shifter, with a 7-speed shifter and with a 6-speed shifter.
The rules of cassettes working with rear derailleurs are:
- The capacity of the rear derailleur should be big enough. Capacity is calculated as: big chainring - small chainring + big sprocket - small sprocket. Example: 28T/38T/48T chainrings, 36T big sprocket, 11T little sprocket, so 48-28+36-11 = 45 tooth capacity required.
- The rear derailleur must support your biggest sprocket. So if you have a racing-style rear derailleur intended for 25 tooth max sprocket, a 36 tooth sprocket will not work. An Acera is not a racing style rear deraileur so this probably doesn't apply. It might be the case the officially supported max sprocket is 32T or 34T, but the difference is so small you could get away with it by adjusting the B tension screw.
To find biggest supported sprocket and capacity of your rear derailleur, you need to find its exact model number. There are plenty of Acera rear derailleurs from different years.
All Shimano 9 speed systems have same sprocket spacing so any 9 speed shifter will work with any 9 speed cassette, no matter if Altus, Acera or something else.
Oh, and an addendum: the chain needs to be the longer, the bigger the big sprocket is. So a new chain will be required (also if going from 7 speed system to 9 speed system this should be obvious anyway).
A second addendum after improving the question:
- RD-T3000 has 45 tooth capacity so it will work with 11-36 tooth cassette if having 28/38/48T chainrings. It will not work (or might be barely at the edge of working and non-working as the capacity specifications might be conservative) if having 26/36/48T chainrings.
- RD-T3000 supports only 34T sprockets. So 36T sprocket is officially a bit too big. The difference between 34T and 36T is anyway so small that I'd say 99% certainly it works well enough.