To my knowledge, there is no way to do that directly with mapserver. I have not tested this, but given you are using postgis you can probably get something close to what you want by creating a POINT layer and a specially crafted postgis query to extract the first point of each linestring. This would be something like:
data "start_point from (select ST_StartPoint(the_geom) as start_point, id, label_column from mytable) as foo using unique id using srid=XXXX"
#color, font, size, etc...
Note that in this case your label will not be oriented along the linestring, and that you will have to render the actual linestrings in a separate layer. You might be able to compute the angle to orient your label inside the postgis query, and bind that for mapserver:
data "start_point from ( select ...,
-- some postgis/sql code to compute an angle in [0..360] from the first and second point -- as orientation,
) as foo ...."