use CGI::Carp qw(fatalsToBrowser); will cause (most) fatal errors to appear in your browser window instead of in the error log.
carpout as mentioned above by Limbic~Region will redirect errors to another log.
If you can communicate to the hosting people, you might be able to persuade them to change the apache setting LogLevel to Warn or better. Then the errors should appear in regular error log.
Comment on Re: [slightly OT] Getting errors and bypassing apache error log
matija,
I debated on mentioning fatalsToBrowser as well. The reason why I suggested carpout extends beyond redirecting errors to another readable log. It is suggested to be put inside a BEGIN block. This allows seeing compile time errors which would normally otherwise be inaccessible. It also allows you to see errors that happen when some other user is accessing it. It helps tracking down problems to have your very own web log when your provider doesn't give you access.
I should have been complete however so thanks for filling in the blanks.