You just need to create the temp table in the stored procedure and then use the output...into temp table. Like so:

create table #TempTable (same strucutre as postcodes?)

with cte as

( SELECT TOP 50 * FROM postcodes TABLESAMPLE (0.1 percent) WITH (ROWLOCK, UPDLOCK)

where Streets is null ORDER BY newid())

update cte SET streets = 'Picked up'

output inserted.*

into #tempTable

I would also get in the habit of listing out your column names and using 2-part naming for your objects.