The grasses chance of long-term survival were nixed at the moment when the seed were planted in the pot.
Compare your pot of wheat, oat or other grains1 to a grain field: The individual plants need a lot more space to grow to their full height. The grains for cat grass are planted very close to each other (intentional overcrowding), allowing them to grow only for a short while befor the natural process of elimination leads to their death. This is intentional because a) that's the only way to get a nice "full" pot instead of a few meager stalks and b) because the customer has to come back and buy a new pot occasionally. Without assuming "unethical" motives, the growers know that most cats prefer soft (=young) leaves over old, tough ones and usually just nibble off the ends.
If you want to grow your own grass, you can plant the grains somewhat sparser than in commercial pots (but not too sparsely) in well-draining potting soil. A rather shallow container (2-4 in / 5-10 cm) will do as the roots won't go too deep. Let grow in a light place, possibly even outdoors until the desired lenght is reached. Do not make the pot available to your cat before the plants have grown to 2 inces at least because the roots should intertwine or your cat may pull the plants out of the pot, making a major mess... Ask me, how I know.
When the blades have been nibbeled on and the pot starts to look well-chewed, you can put it back outside and "mow the lawn", meaning you can cut back the grass to about 1-1/2 inch and let it grow back. If you are really lucky, you may repeat this a second time.
By rotating two or three pots in various stages of growth and re-gowth it should be easy to have some available for your cat at all times.
I personally choose not to fertilize my cat grass because I don't want my cat to eat any residue, even if it's organic and because I know that I will be "using" the pot only for a limited time.
1 I'm explicitly talking about grain-based "cat grass" only because I have read reports of some kinds of Cyperus (also sold as cat grass) causing cuts and other injuries. Better safe than sorry, IMHO.