I agree with other plastic surgeons who have answered this question. I would recommend starting with a hyaluronic acid filler such as Juvaderm or Restylane. It is important to fill the central part of the lip mostly to maintain the lobby of the nature of the central part of the lip. The sides of the lip can be filled to a lesser extent. This will maintain the natural appearance of the upper lip and avoid the over filled appearance with lots of detail. The advantage of this is that it is reasonably priced and although it doesn't last permanently it does last 6 to 8 months at least and it will give you an idea if this is what you want to do permanently.
A permanent result can be accomplished with fat grafting if you wish after that. Fat grafting can be done under local anesthesia in the office. Fat is harvested from a suitable location and prepared and then injected with small cannulae into the lip. You should be aware though that this technique causes considerable bruising and swelling and it may take a few weeks before this resolves.