A hypodermic syringe is "just what the doctor ordered"  for making a fire piston!

I was able to get some 6ml syringes from a farmer's supply store.  The inside diameter of the syringe is just under 1/2" and the seal on the plunger is almost exactly 1/2"

First, I wouldn't suggest using just the syringe as a fire piston. Doing so might be dangerous as the syringe will likely shatter and you could get hurt.

I tired this.  I inserted a screw into the end of the cylinder to seal it off.

I then tried the contraption, and the plunger shattered.  Luckily I was wearing gloves - from which I have to extract one of the fragments!

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I also tried using the syringe cylinder with a wooden plunger and the cylinder blew apart.  Clearly syringes were not designed for the pressures involved in the function of a fire piston

Pull the plunger out of the syringe and remove the rubber seal.

The seal conveniently already has a hole in it.

Drive a small screw through the hole in the plunger seal.  Make a small pilot hole in the end of the wooden dowel.  Assemble the two pieces and you're done.  Note - with the syringe gasket as a seal, the diameter of the dowel is less critical.  7/16" dowel will work just fine and will ensure that you don't need to reduce the diameter.  This saves an extra step.

I don't know how much variation there is in the sizes of the syringe rubber seals, but with the syringes that I have, the seal is perfect, and requires no fiddling to make it work.
 

Now you should have a cylinder, and one or two plungers.

(Note that the plunger with the syringe seal also has an O-ring groove.  This grove is of course unnecessary.  I just happened to attach the syringe seal to a piece of dowel that had previously been used with an O-ring.)

This is what you should have ended up with.

Review the two videos above to get a feel for the amount of friction you should have in your assembly.

Also, note the speed and force that is used to power the plunger.

The critical action is to generate as much speed and pressure as quickly as possible.  It isn't all that critical how quickly the plunger is removed.  Most tinders will stay aglow for many seconds.

You will need some tinder to place into the end of the plunger.  My favorite is tinder fungus.  This isn't always so easy to come by.  Char is a great alternative that is readily made.  Have a look at some of the pages on tinder and char on this website.

*** A word of caution  ***

Please be careful about the types of materials you put into the fire piston.  Any highly flammable or volatile substance can easily cause the fire piston to explode.  Be safe.