Hi Edward,
Yes you can add the Open Cycle Map to ArcGIS Online, you just need to alter the URL a little bit.
From the provided example in Thunderforest's page, the url is like this:
https://{s}.tile.thunderforest.com/cycle/{z}/{x}/{y}.png?apikey=<insert-your-apikey-here>
But for ArcGIS Online, you need something like this:
http://{subDomain}.tile.opencyclemap.org/cycle/{level}/{col}/{row}.png
You can follow the steps here to add a tile layer to the map: Add layers—ArcGIS Online Help | ArcGIS
When you input the URL, you should input this:
https://{subDomain}.tile.thunderforest.com/cycle/{level}/{col}/{row}.png?apikey=<insert-your-apikey-here>
Then you need to define the subDomains - in Thunderforest's case, it should be "a,b,c".
Then you'll be all set and able to use the base map. Enjoy!