You can’t really debug server errors without having really good access to the server itself. This would entail looking at the internal working of this particular host which I’m certain they would not allow nor spend any actual time trying to fix. Move to a better host if they cannot give you more details as to what the problem is or ask for access to the apache and sql logs.
The best you can do really is disable plugins one by one, start with ones that might need lots of cpu like xml sitemaps, or anything running dynamically by a request or action.
Enable slow query logging to pinpoint database problems.
You can also try this plugin, not sure how useful it is, TPC! Memory Usage