How to add link in ArcGIS Online web map popup to google or bing driving directions?

Good question. You can achieve this in an ArcGIS Web Map by providing a custom value in the popup window.

I added a sample New Zealand schools layer by searching for "school" within ArcGIS Online. This sample dataset contains a field called ADDRESS which lists the school's address.

Next choose > Configure Popup:

Choose a Custom Attribute Display, and use the format {ADDRESS} to display the actual value of the address when you click on a point, eg:

The address is {ADDRESS}

Use the hyperlink button on the formatting toolbar, and enter the URL for a Google or Bing search, followed by the address parameter. eg:

https://www.google.com.au/maps/place/{ADDRESS}
http://www.bing.com/maps/default.aspx?Rtp=~adr.{ADDRESS}

Choose Save and OK to close all the dialogs, then try clicking on a school. You should be shown the actual address with a hyperlink - clicking this should open Google/Bing Maps at that location.

See this sample map that I made to illustrate the above. I switched off the default Esri "driving directions" link by unchecking that option within > ArcGIS.com > My Content.