I prefer to store my line on a foam donut. These are easily and cheaply made using toy swimming pool "noodles". the Noodles can be bought for a couple of bucks. Just cut off a slice of foam, then cut a slit around the entire diameter.
When the donut is slid on your tenkara rod, you have an easy to transport package that is ready to fish. The tension from the line keeps the rod sections closed, eliminating the need to place a cap in the collapsed rod. Just slide the donut off the rod, uncoil the line, extend the rod and you are ready to fish.
DO NOT use excessive force when opening your Tenkara Rod to "seat" the sections. Open your rod until slight resistance is felt, and move on to the next section. If you happen to apply a bit too much force when opening your rod, it may be difficult to collapse your rod after a day fishing. It is even more difficult when you are trying to collapse your hands after a day astream with cold, wet hands.
Using a small rubber patch helps you to get a more secure grip on the stubborn section (make sure you only apply force straight in, apply side force may break the rod. One good source for a rubber gripper is kitchen shelf liner. You only need a small 3x3" square so hopefully ou have a partial roll left over from the last time you lined your shelves.
Another good source is rubber "jar openers". These are often given away as promotional items. If you happen upon a source, grab a few and add them to your fishing kit.
Tenkara rods are handy in the fact that they collapse and form their own protective case, at least for light duty wear and tear. If you want to provide even more protection for your rod, it is easy and cheap to make a Tenkara Rod Case.
PVC makes a very economical case, that is extremely strong. A trip to the hardware store, 5 or 6 dollars, and about 15 minutes time will make a very functional case
These cases are extremely strong, you can literally run them over with a car. The one down side, they are a bit on the heavy side.
Update - In the past, I had been making my rod cases out of 1 1/2" thick walled PVC. There are extremely strong, but weigh in at a hefty 1 lb 4.7 ounces. I recently started using 1 1/4" thin wall PVC, and the weight drops considerably, they weigh in a 9.2 ounces. Probably are not quite as strong, but plenty sturdy for general use,