you don't say what time you'll be arriving. From the bus route map, it looks like bus route 50 passes by Philoxenia Hotel. The last bus in is at 6:35pm though.
http://www.nicosiabuses.com.cy/routes.html
Philoxenia is off Aglantzias Ave which actually does turn into the (old) Larnaca Road further down. It's close to the Hilton Hotel (as opposed to the Hilton Park Hotel). Philoxenia is not an actual working hotel at the moment so I imagine that's why you're having difficulty finding the location of it! (It is on the bus route map though!)
I would think that it would take about 40 mins to walk to downtown.
I agree with CBBB to take a taxi. The Kapnos Airport Shuttle webpage (
www.kapnosairportshuttle.com) claims that "Taxis are always available at Philoxenia for onward passenger transportation". The two times I've dropped someone off, there have been a couple of taxis waiting (but this was about midnight) and I'm sure that if there aren't any waiting, they can call you one.
The taxi from Philoxenia to Eleftherias Sq should definitely be under 10Euros - I'm thinking between 8 and 9E. All taxis are metered but if it makes you feel better, you can ask/check that the meter is running.