How to Map a Master Page Gallery in SharePoint 2013 — Office 365 — Design Manager — Master Page
1.1) Open IE and open your Office 365 site and select the “keep me signed in” checkbox while logging.
1.2) Inside your site go to Settings > Design Manager > Upload Design Files and copy the location in a clipboard (for eg. https://domain.sharepoint.com/sites/site1/_catalogs/masterpage/)
a) The Design Manager: Upload Design Files page contains the location of the Master Page Gallery.
b) The location probably ends in /_catalogs/masterpage/. This is the location to which you will map a network drive.
2.1) Open IE in a new window -> Go to ‘Internet Options’
2.2) Select ‘Security’ tab -> Select ‘Internet’ Zone.
2.3) UnCheck the box for ‘Enable Protected Mode’
2.4) Make sure the ‘Web Client’ service is started on the client machine
2.5) In IE ‘Trusted Sites” zone add your Office 365 link for eg: https://domain.sharepoint.com/
3.1) Open windows explorer (I am using Windows 7 Enterprise)
3.2) Right click and select “Map Network Drive”
3.3) Paste your link and select the “reconnect at logon” option:
a) office 365 location link from the clipboard in step 1.2 (for eg:https://domain.sharepoint.com/sites/yoursitename/_catalogs/masterpage/ )
b) you can also try \\domain.sharepoint.com@SSL\DavWWWRoot\sites\yoursitename\_catalogs\masterpage
This should map your drive to the Masters catalog folder in your windows explorer.
Happy Map Drive to your office 365 site!!