Without actually seeing and feeling the linen it would be difficult to know how durable it is. Perhaps you would want to give us the name of the manufacturer and even post a photo of what you are working with.

You need to put in lots of staples. Give yourself plenty of extra fabric, it will tear more at the edges. You can sew a tape to some of the edges (ie everywhere but the corners) to stabilize it.

However, the suitability of the cloth for your chairs is a matter of the sturdiness of the cloth, not the fiber. You can have tough, thick, tightly woven linen and fine handkerchief linen. Ditto cotton and synthetics.

It's called hop sack linen. Here is a photo.

Cut a piece of thin neutral colored cotton 1/2" bigger than your linen on all sides. Now put it over the seat and turn the 1/2" cotton flap back over the linen before you staple. That should give it enough reinforcement so that it will hold. You might want to iron the flap in place so you won't be struggling with it while trying to staple. The cotton fabric underneath will help it if the weave is too loose on the linen.

The fabric you're showing actually looks more like dress-weight linen than drapery-weight linen. You will need to back it as leelee said above, or cut your losses and go back to the store and ask the clerks to help you find a more suitable fabric.

When you're finished be sure to get a spray on fabric protector -- similar to Scotchguard.