I have 2 questions
1. This happens only for GTA 1 or for other games too ?
2. This happened after you installed the game or it worked fine until one day ?
Scan you PC incase you have viruses, or uninstall the game ( and then reinstall it)
If it doesn't work then maybe there are some problems in the registries ... You can use RegCleaner ( I think you can download it for free on any site , just google it ) to clean up the GTA 1 Registry, this way is like you never installed the game onto your machine ... I had the same problem with FIFA 05 ...
Works like this
1.You start the program ( after installing it )
2.Check the list for GTA 1, select it and then press the Remove button
3. Uninstall the game
4. Reinstall it
Dunno if it works 100%
I'm sorry if I didn't helped ... At least I can try ...
Edit: I cheked out the net and found out there is a program that works for any game ( maybe ), it's called SPEED GEAR
Dunno how or if works ...