It depends if your PSU has the raw wattage to handle that, or if the card even needs external powering.

For example, I have a Radeon PRO WX 3100 that doesn't, but it is also a very exceptional card from the norm of gold plated graphics cards that beam the graphics directly into your eyes.

Aaaaand there goes my 'dangerous laughter level' crumple zone.

But anyway, the 3060 Ti is compatible with your motherboard. Nvidia recommends at least 600W PSU for a dependable power supply to the card. Your CPU is no beast, so that's more than enough.

Short answer: all of the RTX 3000 series cards will be compatible with your current motherboard.

More detailed answer:

Nearly all modern graphics cards use the PCI Express (PCIe) interface. PCIe is both backward and forward compatible, meaning that you can mix-and-match motherboards and graphics cards from different generations and they will work together -- though not necessarily at full speed. So if -- for example -- you were using a PCIe 2.0 motherboard with a PCIe 3.0 graphics card, the graphics card would work but it will fall back to the slower PCIe 2.0 speed of your motherboard.

In your case, your motherboard supports PCIe 3.0 and the GeForce RTX 3000 cards are all PCIe 3.0 as well. So that's a perfect match: everything will be running at full speed.

Also keep in mind that even the top-tier of the current generation of graphics cards cannot fully saturate the PCIe 3.0 bandwidth. So, even though PCIe 4.0 is a thing, it's kind of a non-issue for graphics. The current "best in class" GPUs still can't fully utilize PCIe 3.0, and even the next gen of GPUs is unlikely to truly "need" PCIe 4.0.

As other have said, your system is not that old. Certainly not old enough for you to worry about compatibility with newer GPUs at this point. Only thing you need to worry about is your power supply, but if memory serves the RX 580 was reasonably power hungry anyway - on a comparable level with a 3060/3060 Ti, so you're most likely fine on that front too.

Thank you all for your replies. I have a 500W PSU, so I think it might be a good idea to upgrade that too, just in case. Damn, really apreciated the more detailed answer. When you say "interface", you're talking about the "plug area" that connects both motherboard and graphics card? Sorry if I'm making a stupid question :P but really understood everything, thank you!

Yep, exactly.

Yes - I'd recommend a new PSU. The one thing to bear in mind with replacing the PSU is that while messing with PCs is fairly easy, it can be one of the harder bits to replace - there will be wires going all over the place - into the main board, the GPU, potentially to some case fans (depending on how it's all designed). In more compact motherboards, the CPU heatsync and fan setup can overhang the power socket on the board.

I'd recommend that you get a "modular" power supply as you have a small motherboard (and presumably case) - I'd guess most are modular now, but the advantage of these over a traditional power supply is that it doesn't come with the cables hard fixed into the PSU - you only put the ones in you need. In a space constrained build, this will make things easier and encourage better airflow.

If you want to replace your PSU, go for it but IMHO you don't need to. If your 500W PSU is as new as the rest of your system and from a reputable brand, you shouldn't need to replace it.
Cheap generic brands put 500W sticker on every PSU, even when they are 250W. There are some industry standards but most brands don't follow them, each brand has it's own way to market the "Wattage".
The RTX 3060ti should use roughly the same power as the AMD Rx 580, so if your PSU has a max rated Amps on the 12V rail of at least 35A your good to go (look at you PSU sticker), since all the parts in your system are very energy efficient.

Does the RTX 3060ti use the same proprietary nVidia power connector? If it does and the GPU doesn't ship with some sort of adapter you may need a new PSU with correct connectors.

Official requirements on the RTX 3060Ti seems to be a 600W PSU vs a 450W or 500W for the RX580. This probably has some leeway in it for poor quality power supplies, so a good 500W might work. Personally, I'd go slightly higher if OP has the budget as it would make it easier to diagnose issues if there's something wrong with the upgrade. That being said, it's unlikely that a power supply not outputting enough would cause critical hardware failure, so could risk it.

Yeah, I'm kinda concerned about the wires and where everything goes. I was thinking about not changing the way the cables are organized.. not even replacing them. If possible, I just want to unplug them, replace with a better PSU that have the same type of cables (?) and plug them back again. Don't know if this is recommended though.. my PC was bought on february last year, so maybe the cables don't need replacement? I have to deal with my laziness everyday, sorry :P

The marketed max power means very little on the ATX PSU industry, one brand 600W might be the same as next brand 450 or 500W. Is just one easy number to remember and to compare prices.

Sorry for the rant but this triggers me a little :)

From the PSU? You can't possibly mix it up. You can't (realistically) force wrong connectors into sockets that aren't meant for them. Just take a picture of how everything is connected and then redo the wiring. You shouldn't back away from a good learning experience, plus there are three to four billion tutorials on youtube and whatnot. (note that some wires go directly from the case - you don't need to touch them at all, and in fact those are the only 'tricky' ones to figure out when building a new PC)

Here is a good one from Paul's hardware.

You'll want to check if your current PSU is modular. On a modular PSU, you can "unclip" the cables from the PSU itself - and then clip them into a new modular PSU. The connector is universal.

If you can't remove the cables from the PSU, you're out of luck and will have to detach from the motherboard etc. It's not the hardest thing to do, but as you say, it's not ideal.

Or - do as Titanium says and treat it as a learning experience. On my last build, I managed to forget to power in my case fans (they ran off the PSU), and was wondering why I had high temps (computer is in a small office and it was getting toasty in there!) and the CPU fan was going like a jet engine instead of the big quiet 120mm ones in the case. Fixed that one pretty quick!

No, I understand what you're saying and don't see it as a rant - it's why I said that there's probably some leeway in the requirements for poorer quality power supplies, so a 500W might work. If OP goes with an upgrade of the PSU (which he may or may not need, depending on what the quality of the one in his system is), then as a community, we might have to help him find a good one.

Nice, thank you for the video, I'll definitely watch it! Liked the "take a picture" tip!

Yeah, I'll be looking into it as soon as possible. Probably I'll forget something, but with a little patience it's going to be fine! :)