I am surprised that you are flying into Faro to get to Coimbra, as both Lisbon and Porto Airports are much closer.
Coimbra is between five and six hours by train from Faro via Lisbon.
On landing at Faro Airport you will then need to get a taxi to Faro Town. This should take about 15 minutes and cost circa +-Euro 10:00.
On this basis, by the time you have collected your bags, cleared formalities, etc., it is unlikely you will be in Faro until circa 20:15 or 20:30.
The last train of the day North leaves Faro at 19:03 - i.e. before your plane lands. However, this train would,anyway, miss the connecting train from Lisbon to Coimbra, the last departure from Lisbon Oriente being at 21:39.
If you intend to travel by train you will need to overnight in Faro and get the first departure the next morning.
This leaves Faro at 06:55 and arrives Coimbra-B Station at 11:46. This is a direct train and one does not need to change trains in Lisbon.
There is a daily long distance bus service operated by Rede Expressos leaving from Faro Bus Station, 100 metres from Faro Train Station, via Lisbon Sete Rios Bus Station.:
Dept: Faro 20:30
Dep: Lisbon 00:15
Arr: Coimbra 02:50
If you miss this one there is a further departure as follows:
Dept: Faro 01:30
Dep: Lisbon 07:00
Arr: Coimbra 09:20
These are through connecting services operated by Rede Expressos - http://www.rede-expressos.pt/.
You can find links to and/or directly download train and bus timetables pdf format in the "Algarve & Portugal - Travel Transport Links Panel" on the right hand side of my blog at
http://barlaventostravelblog.blogspot.com
