In order to use the Cite Them Right Harvard style in EndNote web, you will need to have your desktop EndNote library linked to your online account.  So when you originally set up your EndNote online account by clicking on 'Edit' then 'Preferences' and 'Sync' in EndNote desktop, any Newcastle customised styles that you have access to should be available to you in the online version too.

To change your bibliographic style in EndNote online, click on 'Format' then 'Bibliography' and on the drop down list 'Bibliographic style' you should be able to find and select Cite Them Right-Harvard.  If you are unable to find the style, try accessing your desktop library and online library on campus, select the style and 'favourite' it in EndNote online.