You can use the Traveline Scotland website to plan your travel in Scotland. But it does take a bit of practice, so here's the answer to your question:
1) CityLink 976
Depart 12:45 Oban (Station Road, CityLink Stances)
Arrive 15:45 Glasgow (Buchanan Bus Station)
2) Stagecoach Express X36
Depart 16:30 Glasgow (Buchanan Bus Station)
Arrive 17:45 Ardrossan (Princes Street, Unionist Club)
3) Caledonian MacBrayne 5
Depart 18:00 Ardrossan Harbour
Arrive 18:55 Brodick Pier
This is the ONLY connection. If you don't travel on Sundays, the journey from Glasgow (Central) to Ardrossan is by train.
If you're prepared to hire a car, then a much nicer way to reach Arran is to drive from Oban to Claonaig on the Kintyre Peninsula and then take the Claonaig to Lochranza ferry. This in itself is a beautiful sail and you will pass through some gorgeous coastal scenery on the way. You could probably drive it in a couple of hours, but might want to take much longer.
Check http://www.calmac.co.uk/arran for details.
Given the 1 km distance from Ardrossan town centre to Ardrossan Harbour and the 10 minute advance check in on the ferry, I think TravelSpy's 15 minute connection in Ardrossan would only work for very fast runners.
I agree with ron_w, now you cannot buy your ticket on the boat you are unlikely to make the boat from the main street in Ardrossan in 15 minutes (assuming the bus was on time !).
I quite agree that if you hired a car the drive from Oban via Argyll and Lochranza is much nicer.
Alternately, break your journey in Glasgow and get the train to Ardrossan Harbour (a very short walk to thr boat and no panic).
It isn't strictly necessary to have a car on Arran, the buses run reasonably frequently Monday to Saturday, but if the weather is poor it certainly helps. Best to book accommodation before arriving, especially if on a holiday weekend.