To get to the abandoned hotel, leave Naxos town by the south and follow the road signs to Pyrgaki - Alyki.
At a crossroads you will see the road signs "Alyko" to the right and "Pyrgaki" to the left. Go towards Alyko, it is a dead end, the road is a little sandy but without any driving difficulty.
The murals at the abandoned hotel are very nice.
https://photos.app.goo.gl/r7W1J3F6pGS68y6TA
The area is also great for its large sandy beaches, either north or south of Alyko. Unlike those around Naxos Town there are no paid sunbeds and the shade is also free provided by the tamarisk trees. :))
https://photos.app.goo.gl/qUyTCXR1fD5kc6Wi7
In this area the Paradiso tavern is nice to have a drink at sunset, if the bartender is still the same, he is a cocktail pro :))
https://maps.app.goo.gl/9mKpUTA4wM38hbAv6
In this area there is an excellent restaurant serving excellent typical of Naxos cuisine: Apolafsi (reservation recommended)
http://www.apolafsinaxos.gr/
Also try to go to Apollonas, at the very north of the island. It is a nice little fishing village.
You will not have enough time to see everything in Naxos, I made this map with the villages, beaches, possible activities (hiking) and places to visit. See what might interest you.
https://www.google.com/maps/d/edit?mid=1d7GZMjFODUlB8lf7EPSXQRwu-YXaKxw&usp=sharing