How can I access the Quick Access Toolbar in Excel for

SharePoint does not have a native URL shortener. That doesn't mean you couldn't use a third-party service like bit.ly to achieve shortening, though.

As for your need to have links--where are you trying to make these links? If you're relying on the Hyperlink/Picture field type in a List, your issue is perfectly understandable, as there is a character limit there, but I've found that using a Multiple Lines of Text field to be a good workaround--if you have rich text turned on, hyperlinks made there don't have that character restriction, or at least it's not so high that SharePoint's own URLs don't fit. Also, if storing links in a Page with hyperlinks is acceptable to you, I've found the character limit is similar to what is experienced with Multiple Lines of Text.