I'm very sorry to hear about your budgie's passing...
Before going out, you can leave the radio on for your budgie, the background noise will give him some comfort and also make him more at ease. Covering the cage on three sides with a blanket, leaving only the front part of the cage showing will also help your budgie to feel more safe after the whole stray cat ordeal.
Yes, budgies do grieve their losses. When my Tito passed away, the whole budgie flock went silent for one whole day and it took a while for them to go back to normal, even though he only had one cagemate, all the budgies in the birdroom knew of his passing and paid their respects with utter and complete silence.
You do well in giving him the much needed extra attention and since he doesn't seem to be coping well with his mate's loss and you are away for long periods of time, the very best thing to do is to get him a new friend.
If possible do quarantine the new budgie in a different cage and room from your other budgie, this is done as a precautionary measure to prevent any possible illness to be passed to your current budgie.
I wish you the best of luck with everything and hope your budgie will soon return to his happy self again.
And despite the very sad circumstances, welcome to the forums!