How best to preserve a rare old game disc while continuing to use the software, and does anyone know the value on this disc?

Disc rot is inevitable. Consider it a ticking time bomb.

Original estimates from the 90s of how long CDs would last (100s of years?!) was totally inaccurate. You can get disc rot in just a few years even from factory discs.

Legally you could copy the game and use it for your own purposes but hacking the Xbox 360 is still difficult the last time I checked.

It would be easier to buy a hacked Xbox 360 and use it to rip the game to storage like USB or HDD. Emulation for the Xbox 360 is still in development and not complete last I checked but keep an eye out for it. That is my recommendation for any rare media.

The best thing you could do is put it in climate controlled storage.

Optical discs - Optical discs like CDs, DVDs and video games should be stored in cool, dry environments. Temperatures between 65 - 70 °F with 45 - 50% relative humidity are ideal. Also avoid exposure to direct light, which can cause clouding on disc surfaces. https://www.storedge.com/climate-controlled-storage-facilities

There are some things you could avoid that would help.

Environmental factors include:

Exposure to high temperatures. Heat buildup speeds the degradation process by breaking down the recording layers of discs that employ organic dye compounds.

Excessive humidity. High moisture content that can seep into disc cracks or scratches and foster the growth of bacteria or mold that can affect the readability of the disc by a drive's laser.

Consistent contact with UV (ultraviolet) rays. Sunlight accelerates the aging of the disc by altering the appearance of the recording layers and therefore rendering the disc ineffective.

Dust, dirt and other air pollutants. These foreign particles can accumulate on the media's surface and in between its crack or scratches. They can not only damage the disc, but they can also cloud reading and recording functions. http://www.cd-info.com/archiving/degradation/index.html

From personal experience you can get disc rot on games kept in a case in safe location so following these guidelines is not a guarantee. Nothing is safe. Every form of storage will eventually deteriorate so just make backups of backups if you can.

It's a really big problem for society and The Library of Congress is actively trying to find ways to preserve optical media.