Im in the room with the 2nd piece of staff of chaos, killed the ghosts but when i take the piece it makes the "tututuu" sound, but it doesnt appear in my inventory and theres no dream to proceed the main quest. Am I missing something?

Please clarify a few points:
- You are talking about Arena?
- You are talking about the 2nd piece of the staff of chaos which can be found in the 3rd ("Labyrinthian") dungeon?
- You solved the riddle "What force and strength ..." correctly?
- When you try to take the piece after having solved the riddle, you hear that sound?

Even with these questions answered I'm not sure I can help you. All I know is that I've managed to get that piece when I played Arena, but that's a long time ago. The odd sound speaks against a problem with the game software, in my mind.
Have you tried to reload an older savegame and attempted to get the piece again?

Yes, Im talking about arena, 2nd piece of the staff of chaos, labyrinthian dungeon. I solved the riddle in another dungeon and killed the boss in another, but got both of the keys to 3rd dungeon. Then i solved the last riddle correctly killed the ghosts defending the piece, but couldnt get it in the inventory and couldnt proceed in the main quest.

I red that somebody else ran into to the same kinda bug, but with 3rd piece of the staff. I think this can be only solved with mod or fix or something but its so annoying that i have to start over the game.

Yes, it is. But it shouldn't be necessary. What about older savegames? Ideally recent ones - I'd probably not go back further than to the dungeon entrance. Watch a Youtube playthrough? Use a savegame editor (uesp.net) - the issue is that you have no guarantee that you can solve the problem in this way, and you would have to understand the structure of the savegame first.
Your game version is 1.06 or 1.07? Have you tried to un- and re-install Arena (backing up your savegames first)? If you use the GOG version, try ArenaSetup or vice versa. Are you using any mods? Then try again with an unmodded version.
Let me repeat that the odd sound you hear may point at a hardware issue (but I'm not too familiar with the hardware side).

Im playing the gog.com version which is 1.07 and tried un - reinstalling the game but didnt help. Using keymapping mod but only after the problem. Trying older savegames.

Tried older saves, did'nt help. I checked the save editors but there's no that could put items in my inventory, first time in my life testing save editors so Greywolf1. Could you help?

I'm not very optimistic :-(. Out of my head I can think of 3 reasons:
- Corrupt savegame. But then it would have to be a corruption which had crept in some time ago (affecting older savegames, too), but not affects anything else.
- Something is missing (action, object). Would assume that the game requires you to do or obtain something before it lets you take the piece. Doesn't make a lot of sense for me, as you shouldn't have been able to get to the piece in the first place. But you never know. Or you did things in the wrong order, or took a shortcut to get to the piece?
- A character condition preventing you from taking the piece. For example, I think I read somewhere that you can contract certain diseases in certain places. Don't ask me how any condition could have such an effect.

In case you haven't tried it already, download and install ArenaSetup (https://wiwiki.wiwiland.net, then click "ArenaSetup" on the left hand side of the window). Import one or two savegames from the GOG version and see what happens.

Good luck!

Got it! The bug is only in v. 1.07. Have to play older versions.

Congratulations!
Which version did you use? 1.07 is said to be the CD update of the most recent DOS version 1.06 (which is the one Bethesda released for free) and to not contain additional bug-fixes compared with 1.06, and 1.07 is the underlying version of ArenaSetup.

I used v.1.07. Maybe I should post gog.com about this bug.

You said you used an older version to get the second piece. Which older version did you use?

There is nothing wrong with informing GOG Support (although I'm pretty sure they're monitoring the forums silently). If you do, it would be useful to add a savefile to your report, which can be used to reproduce the problem together with a standard Arena installation based on the GOG version. And whoever runs into the same problem in the future may find your solution helpful. Thanks for posting it!

But don't expect GOG to be able to fix the problem - Arena is freeware, and nobody is touching the source code anymore (which I think is still with Bethesda).

I havent tried the 1.06 version yet, but i found some link from google that discussed about the game breaking bug is only in 1.07 version. Gonna buy the anthology and find out. Wish me luck!

Looks like I misunderstood your earlier post regarding a problem solution. I don't think there is a need to buy the Anthology. The Arena version contained in the Anthology is very likely 1.06, and you can download 1.06 separately and free (search UESP.Net). I guess it's enough to replace the 1.07 Arena folder of the GOG installation by the 1.06 Arena folder (and keep the DOSBOX configuration). And even if the Anthology did contain 1.07, you would gain nothing by replacing GOG 1.07 by Bethesda 1.07.
Do you mind posting the link to the problem description you mentioned (or tell me how you searched for it - I haven't managed to find such a link myself)?