The way I reconcile this, is by realising that life itself is only ever lived in the now. I don't know that humanity is going to go extinct in my lifetime. I care about the world now, which is why I try to live sustainably, and I've gone vegan. I don't drive a car. I divested from fossil fuels. I encourage all my friends to do the same.
But if my life is going to be cut short by an apocalypse, or if I get hit by a bus tomorrow, ruining the now by dwelling in fear is going to rob me of any happiness I might have.
I also take comfort from the realisation of a few things:
1. The world is finally waking up to the reality of climate change to the point that even China and the US are working together to try and stop it
2. Renewable energy and lab grown meat are becoming more affordable meaning that in the next few years they will become cheaper than fossil fuels and animal products causing a major shift in our consumption
3. Improvements in renewable technology seems to follow "Moore's Law" similar to the improvements in computers meaning we could see exponential increases in efficiency and effectiveness
4. The fact the world created not one but multiple effective vaccines against COVID-19 in a year, despite decades previously never finding a vaccine for a coronavirus shows that the world is capable of coming together to find technological solutions on a scale beyond what we normally would imagine possible. This could be true for technologies like carbon capture.
You might go "well that's all just deluding yourself/wishful thinking, you don't know these things will save you from the apocalypse". You're right I don't know these things will save me. But I also don't KNOW that the world is going to end imminently or that I'm going to die a horrible death at a young age due to climate change either. For all I know, I might get into a car crash and die tomorrow, or I could have cancer right now and not know it. Worrying about it when I can't control it is a waste of the short precious life I do have.
I control what I can control, I help those I can help, and I am at peace.