Please don't get a dog for a dog. And not as a reflex action because your dog is a bit depressed.
It can work wonderfully
But, I know people who have done it, and without fail, every one of them has had awful experiences. It's not like YouTube where you carry the puppy in in a wicker basket and Mozart plays in the background and everyone dances around and there's just lots of love. It can go horribly wrong. At worst, we're talking a dead puppy or your older dog traumatised, at best (If it goes wrong) they just don't get on and you have the odd fight on your hands.
I did it when I lost my cocker (lost him at the end of November and had a new puppy by Christmas). It took months for my other two dogs to accept the Golden (and we are literally talking months - up until about August last year). They get on really well now. But it was bloody hard work.
Honestly, my other cocker has never recovered (fully) from the loss of her brother. Having the puppy has helped her and she loves him and curls up in bed with him. But, she'll often take herself off to her place under my bed where she and her brother slept for 8+ years together - and I know it's because she misses him.
Your dog is 13. He's not a young. Just give him lots of attention and fusses - take him to new places and ensure he's pampered. Then, in a few months, if you think he and you might be ready, then consider it.
But please, please don't do it reflexively.