You grip the lip of the jars with your fingertips and the middle knuckle of your thumb which is why nigiri game jars require a larger lip opening than most normal jars.
The most basic exercise to do with these is an exercise that in America is known as "The Farmer's Carry." You hold these by your sides and walk slowly forward. You can alter the exercise by taking lunges forward or circle stepping (transitioning between stances as you go). There are other exercises in which you lift them in various movements as well.
While traditionally the jars are filled with sand, I would recommend water, especially if you are sharing with another person or want to change the weight often. Water is simply easier to deal with. It's inexpensive, easy to clean up in the event of a spill, and available.
Unlike other hojo undo exercise equipment, these are actually hard to recreate. I do not currently own Nigiri Game Jars. Sensei Pytel once found some vases that worked perfectly on clearance at Target, but they have since been discontinued. If any readers find an inexpensive version of this either email me, or post it in the comments section below.