Why does only one hydraulic intake line from the reservoir have a filter on the Arrow II?
Why is it designed like that?

This design combines two advantages:


The system works reliably with one filter instead of two, saving a bit of cost, weight, and complexity
In case of a blocked filter, the gears can still be lowered (thanks @GgD for the hint!).


How it works

Gear down:




  When the flight deck gear selection handle is put in the gear-down
  position, a switch is made that turns on the electric motor in the
  power pack. The motor turns in the direction to rotate the hydraulic
  gear pump so that it pumps fluid to the gear-down side of the
  actuating cylinders. Pump pressure moves the spring-loaded shuttle
  valve to the left to allow fluid to reach all three actuators.
  Restrictors are used in the nose wheel actuator inlet and outlet ports
  to slow down the motion of this lighter gear. While hydraulic fluid is
  pumped to extend the gear, fluid from the upside of the actuators
  returns to the reservoir through the gear-up check valve. When the
  gear reach the down and locked position, pressure builds in the
  gear-down line from the pump and the low-pressure control valve
  unseats to return the fluid to the reservoir. Electric limit switches
  turn off the pump when all three gears are down and locked.


Gear up:




  To raise the gear, the flight deck gear handle is moved to the gear-up
  position. This sends current to the electric motor, which drives the
  hydraulic gear pump in the opposite direction causing fluid to be
  pumped to the gear-up side of the actuators. In this direction, pump
  inlet fluid flows through the filter. Fluid from the pump flows
  thought the gear-up check valve to the gear-up sides of the actuating
  cylinders. As the cylinders begin to move, the pistons release the
  mechanical down locks that hold the gear rigid for ground operations.
  Fluid from the gear-down side of the actuators returns to the
  reservoir through the shuttle valve. When the three gears are fully
  retracted, pressure builds in the system, and a pressure switch is
  opened that cuts power to the electric pump motor. The gear are held
  in the retracted position with hydraulic pressure. If pressure
  declines, the pressure switch closes to run the pump and raise the
  pressure until the pressure switch opens again.


Source: FAA


Aircraft Handbooks &amp; Manuals: FAA-H-8083-31A, Aviation Maintenance Technician Handbook-Airframe Volume 2 


chapter 13: Aircraft Landing Gear Systems, pages 13-18 to 13-20



Hydraulic fluid reservoir

Ideally, a hydraulic system could be a closed circuit without a reservoir. The actuators have two cavities, one to push the piston out, and one two pull the piston in. The total volume is constant, except for the volume of the piston rods.

A practical hydraulic system has to cope with the fluid displacement of the piston rods, leakage, geometric imperfections, and the thermal expansion of both, the equipment and the hydraulic fluid. A reservoir of hydraulic fluid is needed to allow fluid to enter or leave the system. This compensation flow could be very minimal. The fluid also needs to get filtered, because of possible impurities in new oil as well as possible contamination from the hydraulic equipment and conducts.

Shuttle valve

The shuttle valve is switching the routing of hydraulic fluid: 


When the gears are pulled up, the return flow from the actuators goes to the reservoir, and fluid from the reservoir goes through the filter and the right valve to the pump. 
When the gears are pushed down, the system is mostly a closed circuit. There might be some influx of fluid from the right valve, but that would be a small amount (unless the system is leaking badly).


Without the shuttle valve, there would be no point in the circuit where the main stream of fluid goes only one way. In such a design, there would be no place to put a filter out the contamination from within the system.