If you are sort of "locked" into flying from Cyprus (pax originating there) then overflying Egypt, left turn and overflight of Saudi to Dubai. No issues, really no safety matters if you overfly south of the northern Sinai.
I would highly recommend that you not land at Dubai international, but go to Dubai World..OMDW. The service at the FBO is great and the total time to most destinations in the city will be the same, as your taxi and handling will be far faster. And much cheaper!
Of course, again dependent on where in the Emirates you are going, Sharjah could be a Dubai option as well.
I used to go over Lebanon and Syria and Jordan, but now it is just too hot. Nothing has happened, but you don't want to be the first!
Egypt and Saudi have good overflight services and there are no issues with being N reg.
If you are not locked into Cyprus, but simply planned the flight that way, go via Turkey and Iran(!) Sounds scary but seriously, no issues and the Turkish tech stops are always efficient, less expensive than Cyprus and the overflight of Iran is easy and courteous, as well as being shorter distance if coming from N America or Europe.
I make these trips many times a year, actually depending on we have business in Cyprus and know both routes well. Relax and get your overflight permits in order and enjoy the trip!