I downloaded the file using GoG Browser, after confirming destination folder and licence an error appears:

An error occurred while trying to rename a file in destination directory:
C:\Program Files\...

MoveFile failed; code 5.
Access denied.

Any ideas?

Run installer as administrator and/or change destination folder to somewhere outside Program Files.

Thanks for PM's Thiev, however it doesn't solve my problem.
Few more details:
- I got Windows XP and I'm logged as "Owner" (the only user of this computer)
- First time I run the installation, my antivirus system found a virus in some RCT *.tmp files. It called it "Backdoor.Poison.accj", whatever it means. The antivirus program I use is mks_vir 9.0 Although I turn it off and kill 2 of its processes on ctrl+alt+del list, it doesn't help at all, same error appears every time I try to install it.

Try lowering AV's detection level or disabling active scanning. Although shutting down processes should in theory disable it completely.

The easiest approach would be uninstalling antivir, see if you can install RCT and then re-installing it.