I've always been weary of getting my ears syringed by the doctor. Too much pressure and they can rupture your eardrum. So what I've found instead is a special syringe which has a nozzle on the end that squirts the water out in 3 angled jets, so it still flushes your ears out but doesn't spray directly on your eardrum. I forget the brand of it but you can probably find it with a Google search.
But that's the easiest part of getting rid of an earwax plug. First of all you have to soften it. A syringe on its own is not going to do much good. There are many earwax softeners available in the drugstore, but honestly, I don't think any of them are better than good old olive oil. That's how I got rid of my last plug.
I'd been cleaning the outside of my ear with a Q-tip and in a moment of madness decided to shove it in at an angle because I had this idea that there was a plug of wax in there that I might be able to dislodge. Big mistake - I ended up compressing it up against my eardrum and went totally deaf in that ear. It took me 2 or 3 days of softening with olive oil before a good squirt with the 3-jet syringe got it out. The best way to do it is to tip your head, pour a few drops of oil in the ear, then plug it up with cotton wool and leave it 10-20 minutes. Or in the absence of cotton wool, just lie on your side to make sure it doesn't drip out.
There are some circumstances in which a plug will come out without softening however. Many years ago I was swimming in the local pool and felt something in my ear. An enormous dark brown/red plug had come loose, and I think it might have had something to do with the forces of the churning water around my ears as I swam freestyle.