according to national rail,the 12-01 train is from Birmingham new street this is the main north/south station in Birmingham to be on that train you would have to be on the train at birmingham int.at 11.38,if you land at 11.20,you are not going to make it.Trains leave new street for liverpool every hour at 01 past the hour,the lowest price tickets are £18.00 from b/ham int to st helens central (single) but you MUST travel on specific trains to get this fare,i would surgest a more realistic itinery would be-
12.38 Birmingham International arrive b/ham new st 12.48
13.01 dep b/ham new st arrive liverpool lime st 14.43
14.57 dep l/pool lime st arrive st.helens central 15.15.
(or get 13 38 train at b/ham int and add 1 hour to all other times) REMEMBER to get the cheap fare you quote you must get the xx.38 train from the airport.
you dont mention your return flight times,i hope they are the next day as without checking i think you will be lucky tomake the return journey same day after game.
good luck an i hope this helps