If some website in my server get blacklisted, will my website be blacklisted in my cheap web hosting plan ?

YES. your website will be blacklisted and all the email sent from this server will be in spammer folder. It is called bad neighbor. Some website hosting provider do not do anything to stop that bad neighbor but a lot of responsible web hosting providers will remove your spammer neighbor.

It needs some time for your IP address be white-listed.

You can guard against this by using a VPS or dedicated server, or using a third party email service that is separate to your hosting service.