Calculate vapor pressure from volume
You should treat everything as ideal gas. Then you calculate the volume change from the air itself, using ideal gas law and pressure being constant (atmospheric, maybe). Calculate also the number of moles of air in the syringe, it will come handy soon. 

The remaining change in volume is then attributed to the solvent, again using ideal gas law, you put in the given volume, temperature, pressure and obtain number of moles, which turned into vapor. 
Having this, you know number of moles of solvent and number of moles of air, so again assuming everything ideal (Raoult's law), you know the partial pressures (i.e. vapor pressure).

For added fancines, you can take the decrease of the liquid volume into account.

Side note: While using ideal gas and ideal mixture of ideal gases approximation, all quantities are nicely additive, so don't be afraid.