Apaliunas
As is the case with most Hittite deities, there is not that much out there about Apaliunas. He was the protective deity of Wilusa and is mentioned in a treaty. Interesting fun fact, Wilusa is believed to be the city of Troy/Ilios, which is located in modern day Turkey. Apaliunas is assumed to be the same as the Greco-Roman god Apollo, which makes sense both etymologically and mythologically. Apollo was on the side of the Trojans during the Trojan War and Apaliunas is the tutelary deity of the city of Wilusa.
Aside from this, there is not much else out there about him.
Modern Worship
Typically with a deity that has so little information, it would be hard to figure out how to go about worshipping them in the modern day. However, considering the similarities with Apollo (with them possibly being the same god), there's some educated guesses that can be made. His name possibly means "The One of Entrapment" or "Hunter", which would also track with Apollo's "Agraios" ("hunter") epithet. We know from what little is out there that Apaliunas was a protector deity as well. I would say that, as far as strictly historical information, Apaliunas would be a good deity to pray to for protection and a good choice for hunters.
I would suggest doing divination and/or trial and error to find out if different things associated with Apollo have overlap with Apaliunas. Maybe try devoting archery to him, or learning a creative skill of some sort.
Links- Apaliunas, Wikipedia
- In Search of Anatolian Apollo, JSTOR
Here's my page on Apollon, if you're curious.