I have a 13 to 32 tooth 9 speed freewheel. I wanted to upgrade my bike by getting a clutch derailleur on it but couldn't find much 9 speed derailleurs with a clutch that were under my budget of $60. I finally came across the microShift advent which meets all my needs but as I did further investigation I found that its max tooth capacity was 42. I wanted to know if the max tooth capacity needs to be met for it to work properly or if I can just slap it on without any worries. And would I have to buy the matching shifter to go with it to make it work or could I use my current 9 speed shifter with it? Thanks in advance.
Rear derailleur's not only have a max large cog spec, they also have a MINIMUM large cog spec and a minimal tooth count on the high side (small) cog. You state you'd like to continue using your 13-32 9 speed cassette. When on the Microshift website, and filtering for rear derailleur options for a small cog of 13-14 tooth, the options do not include an Advent rear derailleur or a clutched one. Basically they are 6/7 speed rear mechs that can handle a small cog of 13 teeth. Delving further into the website, specifically the clutched rear derailleurs (Advent 9 speed), the specs are rather slim on information but it appears the Advents are designed for max large cog of 42-46 teeth. Based on their absence when filtering options are set to a 13-14 tooth small cog, that too is out of their spec.
That said, it's possible they (Advent 9s clutched rear der) may work ok with your cassette, however, the large 10 tooth difference in your cassette compared to the Advent max large specs, could be a little too much to expect of the Advent to handle without problems.
Regarding the shifter issue, the information I read from Microshift, is that the Advent rear ders are only compatible with Advent shifters. Two other models of 9 speed rear ders are the Mezzo and the Marvo. These are compatible with Shimano mountain shifters, however, none are clutched and each has a minimum small cog of 11-12 teeth. Here's the Microshift website's 9s rear mechs.
You don't mention what the rest of your drivetrain is composed of, but a clutched rear derailleur is really only beneficial in a 1x system in combination with a narrow-wide chainring for the chief purpose of assisting in chain retention. If you have a 2 or 3x front drivetrain, the front derailleur will keep the chain on just fine. A 1x drive train's chain retention can be bolstered by using a chain guide, which is basically a front derailleur cage. Then a clutched derailleur becomes less necessary. On the flip side, a clutched rear derailleur in a 1x system that does not have a narrow-wide chainring is lacking 50% of the optimal set-up and would need a chain guide at least.
Essentially, it's not making a lot of sense based on your information and noting your limited budget, to get a clutched rear derailleur, until you're willing or able to acquire properly spec'd components to go with it. This is especially so if you are currently running a front derailleur.