Many shelters will cover the treatment of a recently adopted cat but it does sound like Minerva likely has an upper respiratory infection, which are quite common in shelter settings since so many cats come and go at such high volume. Illness transmission can be rampant, particularly in the spring and summer months.
Make sure that Minerva continues eating as much as you can get her to take. Calories are very important in a cat who is ill. The liver can suffer the effects within 3 days of not eating well and in a senior cat this can be seen even more quickly if they already have pre-existing organ dysfunction (difficult to know for sure if they did not run bloodwork on her before the spay). I would personally provide pureed baby food in chicken, turkey and similar flavors. Avoid those that contain onion or garlic in the ingredient panel. I would also try liquid KMR (kitten milk replacer) to see if either of these stimulate her appetite. Often when cats are only drinking solid foods are avoided but liquid or slurry-style foods will be accepted. Calories in whatever manner they will take them, so long as they don't cause illness, is typically recommended.
The shelter will likely put her on a course of antibiotics such as clavamox, amoxicillin, doxycycline or azithromycin. You may wish to talk to them about a course of appetite stimulants, as well, to help keep her appetite where it should be while she recovers from this illness.
If you do have other cats in your home, they've likely already been exposed but it would be ideal to separate them anyhow to try to limit illness transmission. This needs to continue 10 days following the total resolution of her symptoms, just in case, lest you wind up treating the same symptoms in the other felines in the home (again, which may already be a concern since they've been exposed).
If she has problems breathing, you can use saline drops in her nostrils every few hours as needed and you can also use steam therapy in the bathroom during a shower (always monitored) to help break up upper respiratory congestion. She may also benefit from OTC antihistamine therapy such as that of chlorpheniramine. You can give 1-4mg to her up to twice daily to help with congestion.
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