Our 16-year old cat was found to be hyperthyroid and we now rub a compounded topical preparation of methimazole on the inside of his ear twice a day. It's helped a lot. An option for some cats -especially younger ones - is to take a 'thyroid cocktail' with radioactive iodine at a special veterinary facility. The effect is to zap enough of the thyroid to cut down excessive levels. Enough thyroid survives, usually, to result in normal levels. I've taken care of people hospitalized for the same thyroid cocktail. That's mostly done on an outpatient basis these days.
Our cat's cycling has not been affected by his thyroid issues.