Funny! I just had the same exact problem a few days ago.
So this is what I did:
First, of course, I mapped all the Word styles to their InDesign counterpart. Then when I placed it, I had a mess of fonts and sizes (even worse than your Arial…). So this is what I did:
I knew that the only local overrides that I want to retain are Bold and Italic. So I quickly did a Find/Change to Find Format: Bold, and Change Format: ‘Bold’ character style. Then I did the same with italics to my ‘Italics’ character style. (I have these two Find/Change queries saved already, so it takes a second to run).
Then when this is done, I select all the text in the frame (Ctrl+A) and hit the ‘Clear Overrides in Selection’ button on the bottom of the Paragraph Styles panel. And wallah! everything was just the way it was supposed to be!
P.S. If you have more local overrides that you want to retain (such as Underline etc.) just do a Find/Change query for that as well.
Hope this helps.