Most straightforward way to update the A1264 firmware

The big question would be "when" you purchased the AirPort Express since very early versions of the device do require Macs running very old operating systems.

About all that you can do is try to set up the Express using a Mac running AirPort Utility or an iPad / iPhone running AirPort Utility. If you see a message that the Express is not compatible with the operating system, you will need a very old Mac or a PC running AirPort Utility 5.6.1 for Windows.......which was designed to run on Windows 7.

How well it will do on a new PC is open to question.

Latest firmware for the AirPort Express is 7.8.1, which is 4+ years out of date at this time, security wise.

The A1264 should be a 1st gen AirPort Express base station, as you mentioned. The very latest version of firmware that it can run is: 7.8 ... but it was more stable with 7.6.9 or 7.6.8, as far as for AirPlay. I'll have to double-check my notes, but 7.8.1 (which actually didn't resolve the AirPlay 2 issues with 7.8) was intended to be used with the AirPort Extremes & Time Capsules, and not the Express base stations.

The version of AirPort Utility for Windows, as Bob has already mentioned, was designed for Windows 7. However, I have used it with 8, 8.1, 10, & now 11 successfully, albeit I have fewer & fewer base stations to work with as Apple has exited the networking hardware business back in early 2018. The bottom line is Apple has not updated it since it was released ... and I don't expect them to do so any time soon, or ever. Just needed to make you aware.