What options are available to go to Vieux Port and return to the airport later in the day? What are the costs and the travel times to and from the airport?
The bus runs every 15-20 minutes and takes around 25 minutes and costs 8.20 euros one way. It runs to St Charles railway station in central Marseille, from where the metro to Vieux-Port takes under 15 minutes and costs 1 euro 50 one way. (Or 13 euros for ten single-journey tickets, if there are many of you.)
Be warned that there are two very long escalators down from the station to the metro. If this is a problem for you, it is more difficult to get to the port. An alternative if you have problems with escalators is to get a different bus from the airport to Euromed-Arenc in the Marseille suburbs, from where you can get a tram to the western end of the Canebiere, near the port. But this will be slower.
I don't know how much a taxi will be, but I suspect it won't be much faster than bus+metro given how bad the traffic in central Marseille is.
Frequent buses run to Marseille’s St. Charles train station (€9, 3/hour, 25 minutes). You can either walk down to the port area. Once there since you only have a day you may want to look into these tour options: Le Petit Train and L’Open Tour’s double-decker buses. Otherwise you can walk around but it is a big city and walking distances may be longer than you are expecting.
Walking down to the port area from the station is doable. Walking back a bit less so - the quickest way is down a spectacular but long flight of open-air steps, and then down the old main street.
The area it takes you through is perfectly acceptable for anyone used to a big city, but may be slightly intimidating if you're from a small town.