Using iMac Mid 2010 with High Sierra v10.13.16 as monitor with Mac Studio with MacOS Monterey

Not in Target Display Mode, no. Your Mac Studio will not support that.

Use your iMac as a display with target display mode - Apple Support

There are third-party solutions such as Luna Display, but they are only suited for uses where there will not be fast motion and where lag time is not an issue, such as for tool palettes for an app. I think you'd be happiest just buying a new monitor.

Regards.

I just noticed where, in your tag line, you say that your iMac is a 21.5" one.

A 21.5" Mid 2010 iMac would have had a resolution of 1920x1080 pixels. 24" 1920x1080 pixel displays are basically lowest-common-denominator these days. You can get basic ones with poor color accuracy for as little as $80, and one with an IPS panel and near-100% coverage of sRGB might set you back, oh, say, $150.

I would suggest treating yourself to a 27" monitor of some sort, because once you do, you won't want to go back.

The Technical Specifications for the 27" 2009 and 2010 iMacs claim that they support DisplayPort input – but the Apple Support document for Target Display Mode says that recent Macs cannot use any iMac as a Target Display.

At least monitor prices have fallen.

When the 27" Late 2009 iMac came out, a Dell 27" 2560x1440 monitor would have cost you anywhere from $1000 (on sale) to $1200 (list price), so that DisplayPort input feature was potentially a big deal. Today you can get basic 27" 4K monitors with IPS panels and 100% or near-100% coverage of sRGB for as little as $300 – $350. Similar 27" 2560x1440 (2.5K) ones for maybe a bit less. $500 – $700 might get you a very nice 4K monitor.

5K monitors are more pricey. The Apple Studio Display goes for $1600+, and the two third-party ones from LG and Samsung seem to go for street prices of around $800 – $900.