Is there any chance that a friend or relative could meet you at Derry airport? The reason I ask this is because there's a daily flight from East Midlands airport. (Depart 1635, arrive 1735). Unfortunately, the last bus which connects for Cookstown leaves Derry town centre at 1741, so you'd either need to have someone collect you or to find a room for the night.
In case that is of any use to you, I'll fill in a few details:
For the random dates I tried (out Friday 5th Jan, back Wednesday 10th Jan) the Ryanair website offered fares of �2.99 out and �0.99 back. With taxes and charges, that comes to a total return fare of �30.34. (Outbound times as above. The return flight leaves at 1800 and lands at East Midlands at 1900).
Even if you did have to find a room for the night, that might be the easiest way to do it. (If you could find a cheap B&B for, say, �30, that would mean your total costs would be about �60 + travel to/from your home to E. Midlands airport + buses/taxis to/from Derry airport to city centre + buses between Derry and Cookstown).
Ryanair:
http://www.ryanair.com/site/EN/
Northern Ireland travel information:
http://www.translink.co.uk/
Derry accommodation. (Top right of page):
http://www.derrytourist.com/
Chris