「あり」is a pretty old style of the flexion of a transitive verb,「ある,」meaning “be,” “exist” and “lie.”
「池のほとりに一本の梅の木あり」means “There is (stands) a plum tree near the pond.”
「ほかならぬ理由あり」 means “There is a certain reason (for doing that)”
You’ll see「訳あり商品」displayed in a store front quite often at supermarkets, department stores and most of retail stores, which means the products with a certain defect such as a small scar, stain, scratch on the suface of a product, or a product nearing the pull date.「訳あり」in this case means there IS a certain reason for selling the product for very low price.
When you're driving in resort places like the Izu Peninsula and the Hokone highlands, you'll see a lot of signs reading "空き部屋あり" in front of hotels and B&Bs, which means "Rooms available."
Also 「何（なん）でもあり」is a very popular wording, which means "Anything is possible / Anything will do."
「そんなのありかよ」means “Yikes, Can it be possible?” Again, 「あり」here denotes “IS it possible, (acceptable, allowable, permissible) on the earth?”