First things first - yes, you can sync your Haystack contacts to your phone and/or Outlook. There are two methods to achieve this.
Method 1 - turn ON "contacts sync"
When you turn on "contacts sync" for your Haystack contacts, you allow your Haystack contacts to sync to your phone (not the other way around).
If you have your Outlook app installed on your phone, and you allowed Outlook to sync your phone contacts into Outlook - your work is done, as your Haystack contacts will sync to your phone's contacts, and then automatically sync from there to your Outlook.
This works the same on both iOS & Android devices. Click the relevant link to learn more on how to turn on "contacts sync" on Android or on iOS.
Method 2 - install CardDAV Plugin for your Outlook on your PC
In some cases such as your company's Mobile Device Management (MDM) software blocking the contacts sync, or if you want your Haystack contacts to sync to your Outlook, but not your phone, you can also sync your Haystack contacts directly with Outlook on your PC.
To do so, please refer to the below article.
Please note: The plugin works on Windows PCs only. There is no current plugin for MacOS. This said, you can install the 'Haystack' app on your Mac and sync it into your Mac's contact details by just using the first method above.