Пришел ответ из МИДа Кипра на мой запрос. Привожу ответ:
Dear Sir,
You are kindly informed that holders of valid double or multiple Schengen visas issued by Schengen Member States do not need to get Cyprus visa to enter Cyprus. Is not necessary to used it before in the Schengen countries.
Please note that the aforesaid does not apply to holders of Turkish and Azeri passports who need a Cyprus visa to enter Cyprus.
Schengen and Consular Affairs Division
Ministry of Foreign Affairs
of the Republic of Cyprus
Если кому надо, перешлю оригинал. Так что "распечатывать" шенгенскую визу до Кипра не нужно 100%.