1. Is a space port out of commission whilst is is being upgraded or can you still refuel and repair there?
Space ports remain fully operational while being refitted unless some of the components are removed during the refit. Unless the design you're refitting the station to has fewer docking bays and construction yards, you need not worry about losing refueling and repair capability.
2. I have a resupply ship now, but it seems very difficult to refuel it for the first time. Is there some special requirement to get the resupply ship fueled up for the first time?
Refuel in what way? Filling the Resupply Ship's fuel tanks can be done in the same way as with any other ship - dock at a planet or station with available fuel and load up. Since Resupply Ships are generally very large vessels and so might have an abnormally large number of fuel cells, it may be the case that your refueling points just don't have enough fuel available to feed it. Filling up the cargo bays to provide fuel to other ships requires that you deploy the resupply ship over a gas source, preferably one which only contains the fuel gas that your fleet uses (if the source has non-fuel gases, that's probably fine as I believe resupply ships will only extract fuel gases), and will take a variable amount of time depending on how many gas extractors and cargo bays your resupply ship carries.