Small Vacuum pump driving circuit
Would that MOSFET do the job?


At first glance, it's a logic level FET capable of being turned on with 1.8V, and rated at 2A continuous. 250mA is down in the noise on all the graphs. It should work just fine. 


  I couldn't be sure if voltage drop on MOSFET might be an issue for feeding 5V motor with 5V power supply?


Rdson is \$&lt;120m\Omega\$ for the conditions of interest. \$V=IR=(0.25A)(0.12\Omega)=30mV\$ No problem.


  How can calculate Rin and Rgs?


A MOSFET gate is high impedance. Rin is for damping oscillations that can occur when the MOSFET gate and trace leading to it create an LC network. Rgs is just a pull down to make sure the MOSFET stays off when the PWM source is undriven (high impedance). A common pair of values that I start with for this circuit is \$100\Omega\$ and \$100k\Omega\$. They'll work for a lot of situations, but there are reasons to change them that are beyond the scope of this question.