£20 is the 'Weekend First' upgrade available on your outward journey on the Sunday.
Returning on the Monday will require an upgrade to a full 1st Class ticket. This is calculated, in this case, as the difference between between the fare paid (£102.90) and the appropriate 1st Class Return (1st Class Off Peak Return £258.70). A difference to pay of £155.80
That would give a total spend of:
£102.90
Original Standard Class Off Peak Return
£20.00
'Weekend First' upgrade on the outward journey
£155.80
Upgrade to 1st Class on the return journey
Total: £278.70
If you intend travelling after 0930 on the return journey then you'd be better off just buying the 1st Class Off Peak Return at £258.70. Or only upgrading on the Sunday journey and returning Standard Class.
Alternatives would be Advance Purchase if you can commit to specific trains, or splitting the tickets.
Based on current Advance Purchase availability (1.44am 06/01/2015) you can travel out from Birmingham for £93.10 on a 1st Class Advance Purchase on most direct trains between 0903 and 1930. The return journey can be had on the Monday for £65.10 1st Class Advance, but only on the 1149 and 1749. Other 1st Advance prices on the Monday range from £93.10 to £228.90.
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If you're getting the only direct train from Birmingham-Darlington on Sunday, (0903), then you can even stay on the same train, and still buy the first class upgrade.
OP is travelling to Durham, and there's an hourly direct service (twice hourly from 1203 to 1830 at xx03 and xx30) on Sunday 18th January.