Cyprus residency provides access to visas for traveling to EU countries. It does not provide access to freely reside or works in any other EU country, for this you need Cyprus citizenship.
How can I work legally in EU countries after I obtain a Cyprus resident permit?
I am from the Middle East and wish to relocate to Cyprus. I have significant savings to qualify for the program, but I would like to continue working at the same time. If not in Cyprus, can I do it through any other EU country?
Obtaining Cyprus residency will not grant you the automatic right to work in Cyprus or in any other European country. Obtaining the residency scheme via fast track will grant you the right to reside in Cyprus but not work and in order to travel or work in other European countries you will need to obtain residency or work permits from that country. Only obtaining the Cyprus passport will be entitled to the rights of being a European citizen.