With our first puppy we made the mistake of coming back downstairs every time he cried, which soon let him know that every time he whined, we'd come running. Disaster!
With our second pup we were more prepared. You simply have to wait it out and completely ignore him. Do not go back into the room once you've left, no matter how much he complains. It's hard, but it works, and in three days (at the most I'd imagine), you'll have a nice quiet pup and a good night's sleep!
The best thing to do at night would be to put a ticking clock and hot water bottle into a cushion cover. This will give the warmth of your puppy's mother and the ticking replaces the sound of her heart beat. Worked with mine.
As for during the day, I think it's one of those things it'll eventually get used to!
Good luck!
If the puppy is used to you, leave him with an old shirt of yours. Sleep in it the night before and the puppy will be comforted by your scent. And create a ritual for when you leave him alone: giving him a treat, putting him away in a kennel, or in a safe room with his toys etc. That way he knows that when he gets the treat, or is placed in the kennel/room, you will be back soon. When you come home, don't over excite him or make it a big deal. Take him outside to do his business and then pet him. My Basset was very co-dependent as a puppy but these tricks worked for her.
Make sure the puppy has something that is their own. i.e. my dog has a blanket in her bed that we put in the car for journeys, so there is something she recognises.
If she is a bitch she may like some soft toys to snuggle.
Eventually she will stop crying. Just give him/her praise when you return and lots of fuss.
If your dog is lonely, putting the radio on in the background may help, or the tv.