You state, "I was aware of the fence being old" and " my dog goes to their yard," which is where your dog was attacked.
On these facts, you would almost certainly be liable for the vet bills, although the neighbors may be responsible for damage to the fence.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
You are liable for the veterinary bills. I would document your request to your landlord to have your fence repaired. (put it in a letter)
This is not to be construed as legal advice. I do not have an attorney client relationship with you.