Is a Radeon HD7850 compatible with an Asrock H81-HDS mobo?

Hey there japie I only just saw your post.
Sorry for the late response.

I'm not familiar with an H81 board but as Eggz said it physically supports the card with a 16XPCI-E expansion slot and it seems you know this because you actually installed the card with no luck... I had a few questions hopefully to narrow down what could be the cause of the issue.
I see that the Antec power supply you have Antec VP350P does have an available pci-e 6 pin power connector present and just for your information the power supply is a little low in wattage as Eggz said but it is an 18a 12v rail which is enough. I myself have ran an HD7850 running it's maximum factory supported overclock on an older Dell e520 on the stock 305w power supply using a two 4 pin molex connector to one 6 pin pci-e connector for almost a year with no issues. My 305w power supply did get swapped up to a 500w unit when I upgraded the CPU as well. The card itself is not a very power hungry card. 18a is a good 12v source for this card overall. It definitely wouldn't cause the card from not working at all.
So I was wondering if you double checked that the power connection of the graphics card was securely fastened. They clip together but I always check that there's a good solid connection at the power slot of the graphics card and it's not somehow loose in some way. You could check this by connecting and disconnecting the pci-e 6 pin power connector a few times and ensuring there's a click and it's fastened completely if you tug it a little. The other good indicator the card is powered properly is watching the GPU's fans to see if they spin up when the power of the PC is turned on. If they aren't spinning at all then there is no power at the card.
Another thing to rule out is your wondering about it maybe having an older bios and that being a problem. An outdated bios shouldn't be the cause of any such problem on such a recently made motherboard. If it has native 16Xpci-e 2.0 slots its going to work running a GPU out of the box but if there is a newer bios it wouldn't hurt to upgrade it... after you get your video card working.

Now immediately my second thought after the power issue was bios settings. Being you built this computer and you said that the on-board graphics worked fine I'm sure you explored your bios already and maybe even installed an OS... or am I presuming too much?

Generally I would use on board graphics to install the OS and then after completing the install and all updates I would then install the GPU of choice and install drivers. On a rare occasion as I myself have encountered this can lead to there being no output from the GPU because the integrated graphics processor has not been disabled and is initiating itself by default. In order to change this behavior you would have to reconnect your display to the on-board graphics output, and there should be normal screen operation once again at this point, enter into the bios and find the on board peripherals or devices or what have you your motherboard manufacturer has placed the IGP control under and set it to disable the IGP or to initate video through PEG or add in card or pci-e slot 1 or what have you first. On such a newish motherboard I wouldn't think this could be an issue but I wouldn't rule it out altogether.

I would hope that maybe you already figured out the problem by now but if not maybe some of this can help you.

Another question I was wondering is if this was a first build for you or are you fairly familiar with computers.
I had a friend I met who had installed a GTX 650 herself and only played WoW and said it didn't do a thing for her performance in the game or with anything on her computer at all. I got to asking her to look at her computer where all the stuff is plugged in on the back and tell me where the monitor was plugged in... turns out she had been running off the integrated graphics the whole time over 4 months or something. Wasn't even plugged into the GPU and her computer continued to function where it detected the connection only so never initiated the GPU at all. I had her un-plug the monitor from the onboard graphics connection and plug into the GPU and restart and it was fixed... couldn't see any improvement! *Slaps forehead*. She did have the card properly installed and plugged into the pci-e power oddly enough. She at least had purchased and attempted to install the card herself and through the power of WoW was willing to learn. LOL

I have had my own blonde moments believe me. But when it comes down to this stuff it's all experience based and weird things can sometimes happen to a seasoned PC builder just the same. So I will always recommend to go through the simple troubleshooting steps first just to see where the issue may lie. It could be a bad GPU in the end. But more often then not it could be related to any number of little mistake that might have got overlooked.

Simple trouble shoot guide for GPU's

-Check power to the card is available working and plugged into the graphics card
-Check the video cable to the monitor is actually attached to an output port of the GPU and not on any of the motherboards native video outputs
-Ensure the monitor still functions properly and completely
-make sure the bios is set to initiate through the GPU not through the IGP

Again hope this could help
-John