Granada is quite hilly city and almost every direction you go in you will have to go up hills. Plaza Nueva is actually on the river Darro. You may not realise that if you had come to your hotel over 100 years ago you would have seen the river running past your hotel. The river was covered over and now the uncovered part of the river starts at the end of Plaza Nueva by the Santa Ana church. So if you want to have a run which does not have a lot of hills then you can just follow the course of the river Darro. Just turn left out of your hotel and follow the river. When you get to the end of Paseo de los Tristes you have two choices. 1. Cross over the bridge and turn left. You will get to the Fuente de Avellano. This would be a good desination for a short run. Or 2. Go up the Cuesta Chapiz and then turn right into Camino de Sacromonte. Once you are in the Camino de Sacromonte the road will follow the river. Just keep going you will eventually end up in the countryside. Granada is actually like a horseshoe. The west of Granada is so hilly and difficult to build on that it has not been developed.
The photo above is what it looks like if you run down the Camino de Sacromonte for long enough.
The next best option is to run some hiking trails.
There is a whole list of them here
These are the most suitable for someone staying in Plaza Nueva.
This simple route starts in Plaza Nueva and goes up to the Silla del Moro and the Parque Periurbano del Generalife
A trip around the old part of Granada which includes the Albaicín and Sacromonte. It is 7km.
Albaicín 3: Plaza Nueva - Mirador San Nicolás - Plaza Bib-Rambla
This short, 3km linear route explores parts of the Alhambra complex that you can access without a ticket.
Get on the minibus that goes round the Albaicín.
(it passes Plaza Nueva every 15 minutes) Come back to where you started. (it is a circular bus)
Then get out and run where you have just been on the bus.