I'm not sure I completely understood what you're trying to do, but the below code should be fairly close to what you're trying.
If I've missed the mark by a few miles, please post the table definitions.
Update tbl_ApplicantAsset SET tbl_ApplicantAsset = 'Home' FROM tbl_ApplicantAsset INNER JOIN
(select TransactionID, min([Order]) as minOrder from tbl_ApplicantAsset where AssetType = N'House') Filtered
On tbl_ApplicantAsset.TransactionID = Filtered.TransactionID AND tbl_ApplicantAsset.[Order] = Filtered.minOrder
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