Assuming you have an Advanced/Info level license, there are two 'near' tools that could do what you want. The Generate Near Table can create a table that shows the feature IDs of your test and near objects and the distance between them (among other options). This tool doesn't modfiy the original data.
In your case, you would run the tool and use Swamps for your
input features and Land Use as the
near features. You need to specify an
output table location, and that's pretty much it. There are other options available but for what you asked you can leave them at default. Run the tool, then open the resulting table.
Now you'll do a couple of joins to give a little more meaning to that table. First right-click on swamps and join the table you created to it using the swamp FID field (or whichever matches) to the table's IN_FID field. The swamps table should now have the attributes in the table, but while it gives you the distance you want and an ID of the nearest land use, that ID is somewhat cryptic and doesn't tell you the type. So now right-click again on swamps and do another join, this time using the NEAR_FID field you added in the first join to the land use FID. Now you should also have the attributes from the land use features that tell you what type of use the nearest polygon is along with any other attributes stored there. You can hide any fields you don't want or need to see and export to a new layer which will have swamps with the desired attributes.
Alternatively you can use the Near tool in much the same way to save yourself one of the above joins. However this tool will actually modify the existing swamps data (not write a new layer) to add the attributes found in the near table from above. You'll still have to join Land Use to Swamps using that NEAR_FID field and export (or add field) to get the Land Use type instead of an FID as an attribute of Swamps.
Note the distance will be in the units of whatever CRS you're working in.