To solve this problem, I approached their customer service. I asked them if there is any way I could gain access to the cache page by using the URL of the page in a query. Unfortunately, it seems there isn't. According to what their customer service told me, that 'arbitrary' number is the inventory ID of that page in the cache at that time. The key phrase is: at that time. So when the MSN bot indexes the page again, the ID would change and hence any link using the old ID would be obsolete.
Hopefully, MSN will come up with a better way of linking to the cache pages, without using this inventory ID in the URL and rather use the actual URL as a value to the cache parameter like Google does.