Here's an alternative that should also work.

Update ta

SET ta.tbl_ApplicantAsset = 'Home'

FROM tbl_ApplicantAsset ta

where ta.tbl_ApplicantAsset = 'House'

and ta.[Order] = ( select min([t2.Order])

    from tbl_ApplicantAsset t2

    where t2.TransactionID = ta.TransactionID

    and t2.AssetType = N'House')