You could certainly use a hydraulic oiler of some kind, but I think that's way overkill for a lathe spindke. The old school way of oiling a lathe spindle bearing is to have an oil sump inside the spindle head, and a splash oiler. You just drain and refill the sump periodically. A more modern hi tech solution might be to use an air tool oiler and spray an oil mist continuously into the head creating positive pressure. This virtually guarantees that no dirt gets past your labyrinth seals. I would probably go with the sump and the splash oiler myself. It should give you some ideas to think about anyway.
PS If any of this looks a little funny, it's because I'm using voice to text on my phone to reply. Not because anything I said was stupid. Lol!
Bob La Londe
http://www.YumaBassMan.com