Are martial arts a good way to get fit? Or should I just go to the gym like everyone else?

I got an A2A on this:
 
I don't know much about the Japanese fitness industry.  I do know that as far as physical fitness is a by-product of martial arts as opposed to the primary concern of any particular martial art.   
 
If you go to the gym the primary purpose is to get in shape.  Most of the exercises and equipment have that as main reason to show up to their facility.
 
By contrast if you study a martial art the primary purpose might be a martial sport, self defense, or in your case cultural immersion.  However, because it is a physical activity as a output you will get some physical fitness benefit.
 
Because physical fitness isn't the primary purpose of a martial art you get a diminished return on your investment when compared to an activity whose primary purpose is to get in shape.  
 
I am martial arts instructor in the United States and by a coincidence hiking is also a hobby for me. I would offer up in my personal experience that martial arts don't really get me in shape for hiking.  Hiking, climbing and hauling backpacks are really the better things for building hiking endurance. 
 
Things that can also help with hiking beyond exercise are learning to distribute weight in a backpack, eating veggies & fruits in small portions and drinking water often.  If you are going up or down an incline walking with your feet point at a 45 degree angle conserves energy and gives stability.  I sometimes go to all fours on an incline because with a backpack or bag it is easier to navigate.  Those small things give me endurance by not making me use excessive energy while hiking.
 
On a side note you took the time to point out that you are petit and female.  If self-.defense isn't on your radar as something important it should be. 
 
I hope you are enjoying your time in Japan
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