Here's the exact quote:
My own accidental cut across the knuckles was a flea-bite. Doctor Livesey patched it up with plaster and pulled my ears for me into the bargain.
Quoted from https://www.gutenberg.org/files/120/120-h/120-h.htm#link2H_4_0029 (chapter 22, "How My Sea Adventure Began")
Wikipedia article doesn't say much about the history of plaster but lists it in "20th-century inventions". Does it mean some specific design, like band-aids with antiseptic?
Wikipedia there seems to refer to the self-adhesive bandage.
"Plaster" as a word for medical bandages in general goes back to the Middle Ages (Etymonline) and derives from the medieval Latin (em)plastrum with the same meaning and ultimately from Greek ἒμπλαστρον (emplastron). From the Latin it was also borrowed into other languages, like Middle German phlastar and plaster (->German Pflaster). Middle French plastre (-> French plâtre). Old Norse's plástr (-> Norwegian plaster, Swedish plåster).
So it's an old and common term for bandages/wound-dressing in general.
Various forms of adhesive "plasters" using fabric or leaves attached by oil, wax, herbs or honey have been around since at least 2200 BC when they were in use in ancient Egypt. In the era described in the book, late 18th century, Doctor Livesey most likely had several kinds of plasters at hand. Returning to Wikipedia, you will find a description of Diachylon, oil and lead oxide boiled into a paste and applied to strips of linen. Other forms of herbal plasters were widespread, including that of Melilot (sweet clover):
"Take of rozin eight pounds, yellow wax four pounds, sheeps suet two pounds; these being melted, add green Melilot cut small, five pounds. Make it into a plaster according to art."
--Nicholas Culpeper, Culpepers complete herbal, 1815 edition (originally published in the 1650s)
An even more likely candidate is Buchans Common wax plaster:
"Take of yellow wax, one pound; white resin, half a pound; mutton suet, three quarters of a pound. Melt them together."
--William Buchan,
Domestic Medicine 2nd edition 1785
The method of action for these plasters was not well understood and the development of the Band-aid is considered to have begun only after discoveries by antiseptic pioneer Dr Joseph Lister in the 1860s and 70s. Listerine, made famous as a mouthwash, was originally developed as a surgical antiseptic. Brooklyn pharmacist Robert Johnson, of what would later become Johnson & Johnson, built an empire around lead oxide-plasters on cotton gauze in individually sealed packages, and a few years later, in 1920, the band-aid - with gauze in the middle of an adhesive strip - was invented by Johnson & Johnson cotton buyer Earle Dickson, supposedly because his wife Josephine was prone to burning or cutting herself in the kitchen.