I grated mine into a brush knot tolerant sized airtight Tupperware like container. Since 95% of my lathers are generated in a bowl it works well for me. Do you exclusively face lather?
I think I recall @jimjo1031 purchasing and filling new deodorant type dispensers. Maybe he will give some guidance or hints.
Try one of these Colgate shave stick holders. I have a few of these and the one in the picture has an arko stick
I did use a prescription pill bottle, pic #1, which I removed the bottom for a push up guide. Then switched to a vintage Gillette shave stick container I got cheap on Ebay. They have a bottom in which you can push up on to keep it in place. It will need some shaving off the sides a little, then when inserting the shave stick, the container will shave the rest of the excess off for a snug fit, pics #2 and #3.
I think there is a twist up container that fits perfectly. I have seen it on a video by geofatboy. So you could find them https://shavenation.com/collections/shaving-sticks/products/empty-twist-up-shaving-soap-container
Mine is easiest of all. Like jimjo1031, I use a bottle like a prescription pill bottle, except mine originally held a couple tubes of Krazy Glue. The bottle is big enough that the whole Arko stick pops into it with no shaving of the stick necessary. I just keep the stick in it and when I want some, pop off the lid, slide out the stick, use it as normal peeling down the paper as required, then put it back in the bottle and put the airtight lid back on until next time. Simple. Also very cheap since I had the bottle kicking around from buying glue.
Since I prefer to bowl lather I just cut up my Arko stick and then smash the pieces together in a small Ziplock storage container.
Some ideas here ---> http://theshaveden.com/forums/threads/shave-sticking-it-diy-shave-sticks-and-other-stickisms.29016/
I used to use a pill bottle, but for traveling I use a snack sized ziplock bag. At home I remove the paper and just set the stick in an old spice mug. I also prefer to rub the side of the stick on my face instead of the small end. It is much more comfortable for me.
FYI https://shavenation.com/collections/shaving-sticks/products/empty-twist-up-shaving-soap-container