Help needed to select proper electric motor for constant drive
I need about 180 degree movement, back and forth in about one second.
  Maximum torque is about 2.2 kg ⋅ cm.
  The motor should be able run at least month without stops


2.2 kg.cm = 0.022 kg.m
180 degrees x 2 in 1 second = 60 RPM 

Power ~= kg.m x rpm - here = 0.022 kg.m x 60 rpm = 1.3 Watts. 

If this is a one-off then a surplus windscreen wiper motor may be ideally suited - complete with built in rotary to 180 degrees conversion. Loading is minimal and while they are not intended for very long life, your duration should be easily met. 

Any motor rated at say 2+ Watts output and able to run continuously should be a good start - gearing and 1/2 circle motion to be added. A battery drill may suffice - output gearbox probably needed, although the power level is so low that they may be able to run at that speed by adjusting the speed control.
These usually have a one or two stage epicyclic gearbox built in. Driving the gearbox from another drill may be a possibility. - or there are numerous small low ratio gearboxes available.  

You can buy DIY/hobbyist motors with gearboxes with more power than you need.
One key factor in longevity may adding simple lubrication.
Metal gears also helps. 

Stepper motors add driving complexity and are usually expensive for a given power.  

Rotary motion can be converted to simple 1/2 circle drive mechanically - look at how some windscreen wiper motors do this with continuous one-direction motor drive. Or you could add reversing switches. 

A rotary to linear converter / gearbox can be made with a rotating threaded rod driving a captive nut.