Is this possible still? If so around how much would it cost for someone to do it?

 

It is possible to start with 10 hp however it's not possible to finish the caves with just 10 hp.
Final hp level depends on method being used
With prayer in more time consuming so more expensive way it'd result in 23-26 hp depending on luck and ranged level
With prayer in no as time consuming method and cheaper way it'd result in approx 32-33 hp
Price depends on starting levels and method being used