What are the implications of vowing to buy a travel ticket for the Sheikh after its price has increased, knowing that the money allocated for the vow has already been spent, and is it obligatory to fulfill it if one is able, and how?

If you win the case and are able to buy the ticket or pay its value, then you must fulfill the vow on the day you win the case, because that is the time the vow becomes obligatory. It is permissible to pay the value, and the value is considered as of the day the obligation arises.