How long would it take a space probe to reach the farthest star in our solar system?
It is on average 149.10 million kilometers from earth.
As @chandramohanPanakkal said there is only one star in our solar system, so it would be the closest and the furthest... However to answer your question:
The space probe that came closest to reaching the sun was Helios 2 in 1976.
It came within 0.29 AU of the sun, which is 43.43 million kilometers, this value doesn't look that close but in terms of space distance it is fairly close!
The launch date of the Helios 2 probe was 1976-01-15 and its closest approach was on 1976-04-17, which is 3 months after launch (due to its incredibly fast speed of 70.22 km/s).
Helios 2 Space Probe:
Image from: http://space.skyrocket.de/doc_sdat/helios.htm
However this is most likely to change with the Solar Probe plus which will hopefully be launched in July 2018, which will fly into the Sun's atmosphere (corona), with a planned first close approach in December 2024, which would be 6 years in terms of how long it would take to reach the Sun (Making 7 flyby's of Venus first).
Solar Probe Plus spacecraft (artist rendering):
Image from: http://www.nasa.gov/topics/solarsystem/sunearthsystem/main/solarprobeplus.html
To conclude the amount of time it can take for a space probe to get to the sun can be anywhere from 3 months to 6 years, it just depends on the speed at which it travels or if there are missions to be done on the way to our star.
Hope I helped :)