This feature has been implemented, as is available in beta builds since version 22.214.171.124.
The achievements button will update and render when there are any changes to achievements while the app is running (realtime changes); in addition, when the app is opened or foregrounded, it will also check and see if any changes have happened while the user was out (we only do this once every hour to save on cell/wifi data). This behavior should match up pretty closely to the desktop behavior.
When some reputation is earned:
When some non-reputation-earning achievement (e.g. badges) occurs:
Also, the look is a bit different on iPad: