Direct bus at 4pm from Edinburgh airport to Dundee on the 17th May.?

There aren't any direct buses or trains from Edinburgh Airport to Dundee. Any journeys will involve at least one change somewhere.

For travelling by bus, the quickest and cheapest option would probably be via Halbeath park and ride, near Dunfermline. This is doable in about 1 hour 40 minutes, depending on what time of day. From Edinburgh Airport, take the Stagecoach Jet 747 service to Halbeath, this runs every 30 minutes. Then you can get a Megabus service from there to Dundee. You can book a ticket for this whole journey, from either Citylink or Megabus. Megabus is usually cheaper, though the ticket will be for a specific journey. Or you can buy a ticket from the bus driver, though this would be more expensive.

Alternatively, you could go via Edinburgh city centre. Take the Airlink bus service 100 to Waverley, then it is a short walk to the bus station on St Andrews Square. The Airlink bus runs every 30 minutes. Or you can get a tram, these are more frequent, but may take a bit longer. Then you can get a Citylink or Megabus service from Edinburgh to Dundee.

Or you can go by bus or tram into Edinburgh, then a train to Dundee. This would be quicker than a bus, but probably a bit more expensive. For this, get the Airlink bus or tram to Haymarket station, where you can catch a train to Dundee.

To check times for any of these services, you could use the Traveline Scotland website. This includes nearly all buses, tram and train services in Scotland. You can search for journeys on a specific date and time.

The Citylink M92 route has one at 4:15 (1615) from Edinburgh Bus Stn St Andrews Sq to Dundee Seagate Bus Station Arr and then next stop at Dundee Seagate Bus Station Dep that has very few stops in between.

However this is not from the Airport bus stop itself and you will have to travel to the St Andrews Square bus station.

You can take the 747 bus from the airport to the St Andrews Sq station in Edinburgh (it departs at 3.55pm, so get there early, if you miss it you're gonna be stuck there for a long time, and will have to resort to using the train), then at the station get on the M92 at 4:15PM (again, don't miss this, you'll be stuck there) as mentioned above to go directly to Dundee.

If you provide more details maybe I can help more (willing to travel by train, for instance?), however this should answer the question.

EDIT: Unfortunately it seems that even by train, you would have to get the bus first (route 100 goes leaves the airport at 4pm and 4:30pm) to get to the train station. - walk from the airport just past the Hilton Hotel, at 4:33pm get on the 100 to Haymarket, from there walk to the Haymarket train station then get the train from platform 2 at 5:04pm, straight to Dundee.