Becoming Italian resident (with a VISA) brings the new resident to becoma an Italian tax-payer. If the new resident maintains income generated in his country of provenance, there must be verified if Italy has a Double Taxation Treaty in force with his Country. The general principle says that the income generated aborad is taxed in the Country where is generated (plus an additional tax in Italy if provided by the relevant Treaty).
In order to obtain a golden VISA in Italy, the minimum required investment amounts to euro 500,000 (to be invested in an Italian start up). Please consider that obtaining an investment VISA means that the applicant must move his primary residency in Italy becoming Italian tax payer.
Italy still not provide for a free self employment VISA so far. If you want to relocate to Italy and work as a professonal, you do need to apply with the Decreto Flussi (Italian Immigration Decree) which is issued once per year and provide what number and for what works will be released VISAs. There are anyway specific type of professionals (highly qualified) that have the possibility to apply without the Decreto Flussi
Italian Regulation does not provide for a certain amount of passive income for applying for obtaining Elective Residency VISA here in Italy. The evaluation about the capacity to live in Italy without working is deemed to the single Italian Consulate where the applicant submits the request.