226 questions about Final Arrangements are often asked to pay the costs to a funeral home before or soon after goods and services are provided, then those costs are reimbursed to them from the estate.
If your mother's estate won't cover the amount of the burial costs, then there may be other sources to look to for help, such as insurance or Veterans Administration payments, or union funeral benefits.
A possible additional twist: The person who arranges the services may be asked to sign a promissory note to the funeral home personally guaranteeing any unpaid portion of the funeral bill if it is not paid when payment was agreed upon. So check to find out whether either or both of you are legally obligated to pay in this way.
Finally, if you and your sibling are the closest surviving relatives and it is not an undue financial hardship for you to pay, it is the general policy that you split the payment as you can agree.