To show the hits count in WebLink 8.2, the two methods are very similar to WebLink 9.

1. (requires Client 8.3 or EARLIER) Log into any Client 8 as the public user, run a search and move the hit counts column (called  'Count') to the second to the left spot, just after name (If you want it to appear first in WebLink like the screenshot shows). You can also set the other columns to whatever you want the public users to see. Once the columns are set up, log out of the Client and open up the Administration Console. Go to Users and Groups in your repository, Users, and open up the public user’s properties. Go to the Attributes tab and look for [Settings]ColumnDisplaySearch. There should be something in there like “N,179;U,70;XR,132;D,81;” (no quotes). The first column is N for name, and the second is U for context hits (XR is relevance and D is indexed but those aren’t relevant for this question) (the numbers are column widths for the Client, just leave the defaults). Here, you just need to make sure you see the U for hits, or add it in a similar format as above if you don't. Then, go to the WebLink host machine and reset IIS. Finally, open up WebLink as the public user and go to My WebLink, search options. Click the Restore Defaults button at the bottom and you should see the Hits column in the picker.
2. (If you have a more recent version of the Client) The other method would be to skip the Client step of setting columns and just copy this “N,179;U,70;”  into the [Settings]ColumnDisplaySearch attribute for the public user. If you want to include more columns, you can look up their column codes on the support site (or just ask me here and I can get you the relevant ones). Also, it seems from your screenshot that some user is showing this column. The easiest method to fix this issue might just be to open up the user properties for that user, copying the [Settings]ColumnDisplaySearch data and pasting that into the attribute for the public user. That would give the public user the same default columns as whichever user was used to take the screenshot.

As always, please respond if any of this is confusing or any further information is needed.