How long does it take for the Dutch embassy in Ghana to process visa refusal appeals? [closed]

I don't have any specific info on Austria and/or embassies in Ghana but generally speaking a formal appeal has to be lodged with some central court or sometimes a specific body within the relevant's country foreign ministry and not directly with the embassy. The refusal notice should contain information about the procedure for the country you applied to.

A successful appeal will also requires some familiarity with the local language and legal system so you are unlikely to mount one without professional advice. It can in any case take a long time, up to one or two years so it rarely makes sense for an issue that can easily be remedied through a fresh application. See also Schengen Visa Refusal: Does it make sense to appeal?

Furthermore, when Schengen countries have representation agreements (e.g. you apply at the Dutch embassy for a trip to Austria), the representing country can sometimes grant visas alone but refusal decisions should always be made in consultation with the destination country. In your case, it means that the Austrian foreign service probably had the final say on your application and I suspect you therefore somehow need to lodge an appeal with an Austrian court, not to the Dutch embassy (but see the comments on why it's difficult to find out exactly whether an appeal is possible).