What hotel do you recommend in Granada near the Alhambra?

If you want a splurge, you can stay the Parador right next to the Alhambra. We tried to book last October and unfortunately couldn’t get a room, but had a wonderful lunch there. The advantage you are at the Alhambra and you couldn’t have more character and charm. The disadvantage is you have to either take a long walk up and down the hill to get to town or take bus/taxi. Here is the link if you are interested. https://mobi.parador.es/en/paradores/parador-de-granada.

We opted for Casa del Aljarife, which is a bed and breakfast in the Albaicin. Tons of character and charm; walking distance to everything; large rooms and wonderful hosts who are a wealth of information. We had a rooftop deck and a view of the Alhambra.
Highly recommend! https://www.casadelaljarife.com/es/

As mentioned, the Parador is excellent, but extremely pricey. Also at the Alhambra complex is the basic, no frills (our room didn’t even have a tv) Hotel America. It was nice for the couple nights we stayed there. Getting into town is easy, as is returning from town. You can also wander the complex (not the palace or generalife gardens) at night after the crowds have gone, or return to your room for a break while walking around during the day. There are also a couple restaurants about a 10 minutes walk away.

We stayed at the Hotel Anacapri. It is a short public bus ride to the Alhambra. The hotel was in the perfect location for accessing the Albaycin and downtown. The staff couldn't have been nicer. And the hotel is so charming-beautiful rooms and spotless.

I would stay at a hotel nearer to the center of Granada, then take a bus or to get to it.

I stayed at Hostal Rodrí and for $50/night had a clean/comfortable room and bathroom with a little Juliette balcony. Not the greatest views but the family who owned the hotel were charming and eager to help.

If you are a senior (I think they use age 55...I cannot remember what they called the special, maybe sunset or something like that?), be sure to check on the Parador's specials (discounted rate plus a huge full breakfast). When we made our booking, the special was not available for our date, but the Parador advised to keep an eye on their website, and promised to discount our reservation if the special later became available for our dates. We did so, and they did so :)
Our room had a little French balcony that overlooked the Generalife.....fabulous location and fabulous room with original charm.

We especially liked being able to pop back to our room during the day to drop jackets, use the restroom, etc. without leaving the Alhambra grounds. For touring town, it is simple walk to the gates, then one can catch the local tram bus.

We enjoyed the Anacapri - the location is perfect - walk just steps into the city to catch the red bus to the Alhambra. Rooms are very clean and beds comfortable. The breakfast room is very nice and relaxing. Our room was small but we had an additional large balcony.