I'd suggest applying it ONLY to the wounds themselves using a Q-tip, & only a little bit at a time. Neosporin is petroleum jelly (aka Vaseline) based, so it will stick to substrate & wipe off in cage, & it will also smear on your snake's nearby scales (causing more scale damage that turns into problematic future sheds)- Vetericyn is far better for snake use. Topical ointments can be reapplied as needed multiple times daily.
Keep in mind though that if your snake has puncture wounds from your cat's nails or teeth, the antiseptic ointment you apply won't really take care of the whole problem. See my post above- https://ball-pythons.net/forums/show...=1#post2754628
No one can tell you about his survival: just as humans have different abilities to withstand & fight off infections, so do snakes. Snakes are VERY stoic: they seldom seem sick until they're dying, so that's why we're recommending you get him proper vet care- being on an antibiotic for this might save his life.