Good god no. Zuckerberg would be a terrible president (less narcissistic and less stupid than the current one, but perhaps just as dangerous), without the appropriate qualifications, experience, vision, talent, or track record (other than a boatload of money, which I guess in this post-Citizens-United world can substitute for anything else). The concept of having a cabal of CEO billionaires with no political or policymaking experience explicitly in charge of the US government not only leads to terrible outcomes in practice, but is un-American in principle.
Zuckerberg should focus his attention on running his business in an ethical and civically minded way (a lot of low hanging fruit here if he wants to critically examine and do something about Facebook’s sometimes negative effects on people’s lives, news media, and on civic institutions in general), and should not try to single-handedly decide the future form of public institutions.
If Zuckerberg wants to put his money to good use as a private citizen, he should donate it to a charity with a successful track record and expert leadership instead of trying to exert personal control in a field he knows nothing about or turn everything into a personal PR stunt, or he should spend his effort on lobbying against unlimited anonymous spending in political campaigns, probably the most civically corrosive recent trend in American politics.