I want him to pass out, not to die...

How about an arrow tipped with poison from the Pufferfish. That paralyzes people for a long time, but doesn't usually kill them. It's what Voodoo practitioners use to create zombies. (The walking dead, not the drink)

Richard

An arrow that cuts an artery or a large vein could cause enough blood loss for the character to pass out. If you want him to lose consciousness quite soon, he will have to lose a lot blood very fast, and will have to get modern medical attention (surgery and a transfusion) pretty fast. If you don't have access to modern medical stuff, it might be more realistic to have the character faint (a vasovagal reaction causing a drop in blood pressure). He wouldn't have to lose much blood at all, just be frightened and in pain, and would recover pretty much right away as soon as he wakes up (a few minutes).

So shoot him someplace with no important organs, like arms, legs, buttocks, shoulder. Slicing off an ear would be impressive and scary, but not very dangerous.

Head wounds bleed rather nicely, often enough to make one pass out briefly... let the arrow skim his scalp (not enter the skull).....

Audrey

I had my character grazed on the arm by the arrow, lose his balance, fall, and hit his head on a rock. He's out, but not for good. The traveling party up the road was not so lucky.

If your char is already dehydrated (say he's been sweating all day, fighting, running or pursuing), then it takes relatively less blood loss for him to have an altered mental status/loss of consciousness due to hypovolemia.

Or if he isn't hardy to pain, he can pass out w/ little blood loss.

Some folks literally cannot handle giving a unit (500cc--about a soda can and a half) even larger people. Small people often cannot give a unit, therefore never donate blood. Other folks might handle losing two or three units, which, splashed about, looks like an awful lot of blood.

"I want you to pass out, not to die ..."

I think my hubby said that to me once ...

Carry on

If your character received a grazing blow to the head there would be enough force to knock him out for as many hours as you need. There would be blood, but not a life threatening amount. The resulting concussion would cause problems for a while, but he'd be mobile and could fight with reduced capacity.

Where to aim?

Anywhere that nothing vital gets a hole poked in it. This is fiction. You get to play like you like (all while keepin it real, man...)

Then there's this: I once watched a guy crash a bicycle while transporting a quart of beer in a glass bottle (don't drink and drive … bicycles included). He landed palm-first on that glass bottle, broke it, and glass cut deeply into that hand, severing tendons, veins, an artery. He came running across the street toward me, great big gouts of dark blood pumping out of that hand, a look on his face like, My God, I'm Going To Die Right Now! Thirty yards later he was in my shop, all weak and pasty white and muttering about everything going black. I sat him down in a chair and clamped off the leak, all the while keeping the wound elevated, telling him in nice, calm words that he'd be okie dokie. Short of that, I'm pretty sure he would have conked out on the spot. Dude was withered, wilted, and weak. But I think it was more from queasiness on his part than blood loss.

So is your character queasy? Pass out at the drop of a hat, stub of toe, prick of finger, sight of needles?

'Nother story: Guy I once met got into a fight with a dude who pulled a knife and sliced his upper arm to gory bits in four or five slashes. Guy puts the dude down and feels all weak-like, just sleepy tired-like, and that simply from blood loss. But then there was another guy I met whose car got hit by a boulder barreling down a mountainside. Car flipped over and crushed his arm mostly off at the biceps. Gory bloody mess, and he never once lost consciousness despite suffering a much worse looking wound.

Based upon that anecdotal evidence, I still say it's pretty much up to you make up what happens, and to give your readers a plausible explanation. That's all. Unless you specifically need to explain medical details, most folks will go along for the ride if you make the scene believable, either by saying that your character passed out due to blood loss, or shock, or both. Easy as words on a page, and the willing suspension of disbelief.