Firstly, you are travelling on a Saturday morning and need to allow yourself at least 30 minutes to get from central Porto to the Airport by Metro. This means you would need to get the 06:15 service from Metro Trinidade as there are only services every 30 minutes at this time on Saturday mornings.
The timetables for the trains between Coimbra and Porto for April are not online yet. However, according to www.cp.pt (Portuguese Railways) the first train at present on a Saturday morning leaves Coimba_B Station at 6h40 and arrives at Porto at 8h35. So this option is obviously not possible.
However, the bus operator Rede-Expressos (http://www.rede-expressos.pt) have a bus leaving Coimbra at 03:00 arriving Porto at 04:25. It would then probably be best to get a taxi to the Airport.
This is the only option that seems to work.
Alternatively, you will have to travel up to Porto the night before and overnight there.
