GG,
I don't know if you have ever traveled to Europe from the States, but the time difference from New York to Cyprus is 7 hours, which means you will have a jet-lag for at least 3 days and maybe even longer. You will find that by late afternoon, you would want to go to sleep and around 3am in the morning, you are ready to get going, but everyone else will be sleeping. So don't count on doing very much in the evenings for the first few days.
Are you really coming all the way from the USA just one week, or do you have other places to visit before or after visiting Cyprus. It will really be a shame if you could not stay longer.??
As GR said, without a car, you are spending too much time chasing buses. Good thing you have few forum members who are willing to show you around.
Have a nice trip.