Waking up early for the drive up to catch sunrise is a MUST DO if you are in Hobart. It's only a twenty minute drive if you're living in the CBD - you can do it. But do check the weather report in advance. Depending on time of year, snow and/or fog can make it extremely difficult to ascend in the morning without the aid of street lamps on the way up.
There is a way to camp atop Mt Wellington to save yourself early drive / hike up - and it depends on the ranger's generosity too (they go on patrol often), but if you read the park notes carefully, there are areas for you to pitch, as long as you are a comfortable distance off the roads.
Of course, mountains as such have their own cloud 'tempers' and you may not always have the right cloud conditions/positions for a great photo. Go here to check out road conditions to the peak.
Nevertheless it is dynamic enough and perfect for time-lapse. 15-minute at 5-second intervals will suffice if you are pressed for time. Otherwise, find a spot for sunset. Some clambering over rock might be required, which really adds to the fun.
Sunrise or sunset atop Mt. Wellington - amazing with the right wind conditions. Don't allocate just one day for this spot and leave disappointed, if you don't get the great views for the first day, come back the second and keep your fingers crossed. When the spectacular river system unveils itself to you from the clouds prepare to have your jaws relax in high definition enjoyment of nature at its finest.
The Organ Pipes, a 20minute walk from the peak of Mt Wellington is also a worthwhile investment of time.