Thank you for asking me.
Since you have just started studying Buddhism, I am sure the first thing you have studied would be the 4 Noble Truths as expounded by the Buddha. The first universal truth is that this world is by its very nature very unsatisfactory. It pervades to the core of everything existing in this world. The ultimate unsatisfactoriness is death. Lurking in every corner is the thief of death. If we were to rise above the world and look down, we will see beings wallowing in the quagmire of sorrow. Beings include every living thing. Read the newspapers, watch the news, watch the National Geographic and Discovery programmes.....Lives being eaten, lives getting killed, some for no valid reason at all. Innocent people, including pitiful babies and children suffer in the hands of humanity and the wrath of nature. Go for a "tour" to Iraq, to Afghanistan, to the warring states in Africa, to the slums of Asia and South America. Then you will realize that your beloved cat had indeed lived a very good life and "died" at a ripe old age.
On a more practical approach, you can always start again the "vicious" cycle of getting a new kitten, and all your miseries will be forgotten in a matter of time. But be forewarned, this kitten will suffer old age, sickness, and death in time to come. This is the Buddha's revelation of the Truth of life. When there is life, there is old age, sickness and ultimate death. If we can realize and accept this unrefutable truth of life, then we can try our best to live in peace in the midst of this "Dukkha"...unsatisfactoriness, sorrow and misery. We can have inner peace because now we understand the nature of life and accept that which are natural and cannot be changed.
This is the real nature of life.