I doubt you would be happy with the result. You'd certainly loose the carbonation and probably oxidize the cider in the process. Angry Orchard is also kind of the Bud Lite of the cider world, it isn't exactly a high quality ingredient to start with.
You don't need an orchard, a press, and lots of shiny stainless steel to make cider at home. Free food grade buckets from a bakery, sanitizing solution (I really like StarSan but there are plenty of alternatives), an auto-siphon, a hydrometer, and an airlock are all you really need to be making cider from preservative free grocery store juice. You can easily get by with less but what I suggested is probably less expensive than an oak cask. If oak flavors are important to you oak chips, oak cubes, oak honeycomb, and whole barrel staves are all available but I'd suggest long term aging in a PET plastic or glass carboy.
