I think it’s important to be aware of the spiritual effects, the effects these systems can have on your whole person. I say this as someone with 5 years of Karate and 2 years of Iaido experience.
While there are elements of Eastern mysticism in some martial arts systems, it has to be said that much of what the Chinese understood as mystical and supernatural is really just a pre-scientific understanding of the natural. Notions of Ch’i for example, while tied into a cosmology that is un-Christian, basically boil down to pragmatic notions of the body’s energy and breathing. If your faith is strong, and you are not too susceptible, I would say you could approach the mystical elements with the proper scepticism of a faithful Catholic and still get some good things out of the physical side of martial arts training.
However, the more serious spiritual effects can come from the adrenaline that comes from fighting. Don’t forget that many of the systems, particularly the ones made popular in the West, such as Judo and Karate, were basically invented in the early 20th century by the neo-fascist imperial government of Japan to train people to be better soldiers, that’s why there’s so much emphasis on discipline, on repetition, etc. The same can be said of many of the more commercialised Kung Fu schools, which have their origins in the Boxer Rebellion and the Chinese Republic. I stopped with Iaido, Japanese sword fighting, because it places so much emphasis on striking to kill - ultimately, as beautiful as it looks, and as much as it teaches precision and control, the only real reason to draw a sword is to kill your opponent, and it doesn’t do the soul any good to constantly re-enact that desire, even if you claim you are learning to control it. Having said this, I’d say the same about making regular visits to a firing range. (When you hold a weapon that can kill - and the sword is both gun and bullet, so you really do feel the power it has - you realise why widespread gun ownership is such a bad idea, but that’s another topic.)
Don’t be fooled even by the idea that there are generic virtues. All the virtues are universally accepted in the broad sense, nobody would say they disagree with bravery for example, but what bravery means can mean different things to different systems of thought. Take truth for example, both Catholics and atheists would say they were committed to an honest search for the truth, but the origins of their notions of truth would be very different.
All I’d say is the same common sense advice about taking up anything that’s going to have an effect on your life, “by their fruits you will know them”, take care and watch what effect the practice of martial arts has on your life, your thoughts, your actions. I’d say the same about watching TV, or reading a series of books, taking up a sport, or a new job, campaigning for a political party or even attending a Bible study. Just watch where it leads you, be ready to walk away if it leads you away from Christ, even a little.
Sorry to have rambled so much, just my take on things, sometimes the ‘Eastern spirituality’ question masks more obvious issues and concerns with the martial arts. I’m not saying they’re good or bad, just saying what I think.