If you're trying to copy object data to a map with NO custom object data, all you do is go to the object editor, Select "file", "Export all object data", and give your object data a name (and remember in what folder you saved it).
Then go to your new map, open object editor, select "file", "import all object data", and select the file you just made 2 seconds ago.
Warning: this will remove all object data in your new map. So if you already created custom units in your new map, these will be overwritten!
If you have custom units in both your old and your new map, the "easiest" way to import the custom units from your old to your new map is:
Open object editor of your Old map, and open the "Campaign Editor". Then open the object tab in your campaign editor. Copy all units from your old map into the campaign editor.
Next step: open your new map, and keep the campaign editor open. Then copy the units in the campaign editor into your new map.
Why is this useful? If you do it this way, you won't have to switch between your new map and your old map for every single unit you try to copy. Could save alot of time.