That will turn off notifications altogether, it should check and display only when its open in multitasking.
I see for Mail you can switch off push so that it doesn't constantly check and saves battery. Is there similar option for individual apps, how do you stop them from using Push?
The only options that iOS provides system-wide is the Notification Centre options. The options here are all or nothing. Show badges for all notifications, or don't show badges, show banners for all notifications or don't show banners, etc.
However, some apps will provide their own internal options for what notifications you want to receive in the first place, or the times or circumstances in which you want to receive notifications (e.g. in Facebook you can mute conversation notifications for an hour, or disable post comment notifications, etc. etc.)
An interesting case story is Skype, who up until a few months ago only pushed notifications to your phone if you were logged in and had the app running. They changed it so that now you receive notifications even when the app is shut. You have to sign out to stop receiving them. I found that change incredibly annoying due to the particular way I use Skype. They provided no option, but it was probably perfectly possible for them to do this.
So what you want to look for is options within WhatsApp to control which notifications you receive, when and under what circumstances. If it does not have these options, i.e. there is no way to say 'don't push notifications if the app is not running' there is not much you can do, as far as I know, other than temporarily disable notifications in the Settings app and/or in WhatsApp's internal settings, when you shut the app down.
In the attached image, you need to turn off the two Alert toggles when you shut down, and turn them back on when open it again.
P.S. Push notifications are different to Mail push. All apps' notifications go through Apple's push server. Only one connection to this server has to be maintained. Allowing many apps to send you push notifications doesn't use any more battery than allowing one. Although, of course, the number of notifications you receive is likely to increase as you allow more apps, and each notification uses a tiny bit more battery than idle. But, say, disabling WhatsApp push notifications overnight won't save you any battery if you don't receive WhatsApp notifications overnight anyway.
Mail on the other hand, will maintain a direct connection to each account's mail server, so turning off push for a Mail account will mean one less connection, which saves battery (vs not fetching at all; I won't get into the argument over whether regular scheduled fetches use more battery than maintaining an idle push connection).