No.
Your Mac does not support the 5K (5120x2880) resolution of the Studio Display (2022). It supports resolutions of up to 3840x2160 pixels.
MacBook Air (11-inch, Early 2015) - Technical Specifications - Apple Support
MacBook Air (13-inch, Early 2015) - Technical Specifications - Apple Support
Apple has a list of which Macs and iPads are compatible with the Studio Display, and your Mac is not on that list.
Studio Display - Technical Specifications - Apple
Using a USB-C to Mini DisplayPort adapter won't generate a Thunderbolt signal even in the unlikely case that the adapter works at all when used in reverse. Spending $80+ on an Apple Thunderbolt 3-to-2 adapter and a TB 1/2 cable is also pointless given that the Mac is too old to support the 5K resolution of the display.