What are the pros and cons of vacuum and pressure gyros?
Inspired by this question, I would like to know what the pro's and
  con's of the mostly used vacuum-gyro and the less used
  high-pressure-gyro is.


It is largely related to the instruments in your aircraft but in reality its all about air pressure and the systems are really doing the same thing in the end of the day. 

Lets look at the evolution a bit, early aircraft that lacked an electrical system generally used a small external venturi to generate low pressure suction for the early gyro instruments. Note the venturi tube in front of the door about half way up the body on this early Cessna 170. 



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Since it was easy to generate low pressure this way early instruments were built as suction based. As planes got faster and external (drag educing) things were removed these systems went internal. Since the instruments were already suction based it was logical to make engine (or electrical) suction pumps to drive them. 

When pumps moved into the engine compartment they came with some issues. Wet suction pumps have an oil issue that can seep into the system and cause issues with the instruments. That was solved with the advent of the dry pneumatic pump however that was a bit more susceptible to debris clogging but could provide both positive and negative pressure. 

This paper outlines both systems as well as their pros and cons in depth and answers the question quite well. However the end result is that both systems provide the same outcome and in the view of the instruments there is no real difference.   


  Gyro Instruments Within the case of the gyro instrument the air flows
  through the air seals of the gimbalsʼ bearings and across the rotor,
  spinning the rotor on its axes. The exhausting airflow from the rotor
  housing applies precession forces to maintain a reference to earth in
  the attitude gyro and the attitude of the aircraft in the directional
  gyro. The mechanics of the gyro instrument remain the same, whether it
  is driven by the airflow in a pressure or vacuum system. The air- flow
  through the instrument is a result of the differential pressure
  between the instrumentʼs air inlet and the air outlet ports. In a
  vacuum system the pressure at the air outlet port is lowered below
  atmospheric pressure. The differential pressure between the atmosphere
  and the pump now forces air to flow through the instrument. In a
  pressure system the pressure at the air inlet port is raised above
  atmospheric pressure and the airflow is in the same direction.


So what are some pros and cons, 

In a vacuum system the filter is upstream and air is pulled through it and it does a good job of keeping the air clean. 


  ...filtering the air entering the instrument. This filter will remove
  most contaminates from the air, but moisture is not one of them.
  Periodic replacement of this filter will ensure clean airflow through
  the instruments.


While in pressure based systems the air filter is generally mounted in line but has a similar effect 


  In a pressure system there is an inline filter in the line between the
  outlet port of the pump and the inlet port of the gyro instrument.
  This restricts, but does not completely stop, the flow of all
  contaminates to the instrument.


However pressure systems have a moisture problem since the compression of air creates moisture, this may add moisture while a suction system will only draw ambient moisture.  


  Unfortunately when air is compressed, moisture is created. Although
  the pressure is extremely low (approximately 2.5 to
  4.5 PSI), this system is no exception. Moisture is the No. 1 contaminate, contaminating the gyro bearings in a pressure system.


The paper also outlines that both systems are susceptible to downtime due to pump inspections and system maintenance. Which is a con of both systems. 


  Side-question: Are there any other ways to make the Artificial Horizon
  work?


Yes, electricity 


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