I'm a moderately experienced DM, and I'm currently running a game for an afterschool club with some classmates.  Last session went well, except for only two people showed up.  I had the cleric's character sheet and I asked him if I could play his character (He ditched us to go play Magic :/).  He said yes.  The party is currently on level two, and without a third party member, since only the one of the wizards and the fighter were there, they would've gotten creamed.  Anyways, they decided to fight a bugbear.  I had the cleric do some melee attacks against it since his character sheet didn't have any spells I could use, and missed every single attack except one.  The other party members did some decent damage against it though, so it was down to 5 hit points and about to die, but then it rolled a crit hit on an attack on the cleric.  And knocked him to 0 hit points.  Whoops.

We haven't done any death saving throws for the cleric yet, because the bugbear died literally one turn after it knocked him out, and then we had to end the session as soon as the fight was over.
Anyways this is my fault and I feel pretty bad about it.  I don't really want his character to die already, especially since it's my fault.  The only problem is that none of the characters have any healing abilities whatsoever.  No healing spells.  No items that could help.  I don't even think anyone has any proficiencies in Medicine.  Except for the cleric.  Anyways, does anyone know how I could fix this?  Should I just have someone roll a Medicine check to stabilize him and hope for the best? Or is there some sort of more interesting way I could solve this narratively?  Thanks.

If a player trusts me (as GM) to play their character, I give them a sort of "soft plot armor" for that session. They typically will fade into the background and only do the minimum. If enemies attack the "AFK" player's character, I will fudge the damage to prevent them from dying. If the "AFK" player's character attacks enemies, I will equally fudge the damage to prevent them from being the "star" of the session.

That doesn't really solve your current situation, but if combat is over and no death saves have been rolled, there really isn't any risk of him dying. Either have an NPC or one of the other PCs do a DC 10 Medicine check and he is stabilized. No harm, no foul.

Oh yeah! I forgot about that part of how death saves work to be honest lol.  His character was in the background the whole session, like you said.  I didn't really think about fudging the numbers though, I'm a bit too honest for my own good sometimes.  Man, I didn't realize that it was only a DC 10 Medicine check for that though, I thought it would be higher...
Anyways, thank you!  Sorry for wasting anyone's time with a sorta pointless question.

I try to avoid fudging dice, but I will if I think the combat becomes "too" easy / difficult for the party or in a scenario similar to how you described. That is why you have a GM Screen. :-)

And don't apologize, that is an important question and there might be someone else he wanted to know but was too afraid to ask.

Yup, easy-ish, medicine check to stabilize the cleric, unless.....

This could be a way to introduce the seriousness of death to the party as well. I don't mean in a drastic, doom and gloom sense, more in the story and game play sense. All too often DMs complain that their players don't understand that combat is not the only answer, so you now have the golden egg right in front of you. Create some sort of mission to save the cleric from death, create an overwhelming sense of urgency, make the tension palpable, and have your players sweat...but win. The cleric's owner needs to agree to this however, it's unfair to do otherwise. You can also use this to create a parallel sub-plot where the cleric is gaining a minor favor from their diety and get a nifty game long buff as an apology for almost killing him :)

If the cleric hasn't made any death saves yet, odds are high they'll survive.

The odds of the cleric stabilizing on their own are almost 60%. The DC for the Medicine check is only 10, so even with a -1 modifier that's still a 50% chance of succeeding. The cleric has two allies, and they each have 2-3 rounds (depending on initiative order and luck). If they get 2 rounds, there's only a 6.25% chance all of their checks will fail; with 3 rounds, the odds go down to about 1.6%. If you want to further improve their odds, you can award the survivors inspiration; there are different ways to rationalize this depending on how the fight played out and their respective personality traits, bonds, and flaws.

If anyone (including the cleric) has a Healer's Kit, this is a non-issue; stabilizing them will automatically succeed.

Some tips for the future:

• Rolling behind a DM screen will let you fudge the dice in situations like this, where a player's character might get killed through no fault of their own. In other words, you could've pretended the bugbear rolled a 19 instead of a 20 and nobody'd know.
• 3 person parties need to be handled with a lot of care; not only are encounters harder for them, it can be difficult for them to recover from 1 party member getting KO'd mid-fight without access to Healing Word.
• Level 1-2 characters also need to be handled with care.
• You can adjust the HP of a monster within the range of its hit dice to suit your needs (a bugbear can have anywhere between 10 and 45 HP.) If a fight isn't going the way you want, you can always decide the monster had less (or more) HP than the average. Even if you don't roll behind a DM screen, this is always an option!
• Remember that you can award inspiration at any time, for a wide variety of reasons. This is useful when a situation hinges on a single die roll and you want them to succeed.
• Encourage your players to get Healer's Kits for situations like this.

I'd suggest just having him start the session stable, but at 0 HP. A few hours and he'll be right as rain