The short answer is no.
The Neon, I believe, has hydraulic brakes and the equivalent of a Tiagra 10 speed drivetrain. Tiagra drop levers don't do hydraulic, so you will at least need to buy 105 levers - 11 speed, so
I think you'll just have a redundant click, unless you also buy a new rear mech, cassette and chain. You will probably need new hydraulic hoses, and will anyway have to do a complete brake bleed. You will probably also need a different stem. This is all getting uneconomic - better to sell the bike and get a drop bar bike with the correct geometry.
A possible alternative is to buy some
bullhorn bars and use your existing levers. Might want to change stem, but no brake bleed and easy to do.