If the above info comes from his discharge papers, it means only that he was assigned to the regiment at the time of his discharge and its anybody's guess as to when (or where) he transferred in. Although organized early in the war, the 87th Infantry Division - of which the 346th was an organic element - didn't go overseas until late 1944, arriving in France just in time for the Battle of the Bulge. The 346th participated in three named campaigns: Rhineland, Ardennes-Alsace and Central Europe. He should be credited with at least one of these if he served with the regiment in Europe.
Re the Aleutian Islands Campaign. All who served within an officially-prescribed area of Alaska between 3 January 1942 and 24 August 1943 received campaign credit. Qualifying individuals may wear a bronze, 3/16' service star on the Asiatic-Pacific Theater Ribbon. All who participated in the assault and seizure of Attu Island, 11-30 May 1943, are entitled to wear a bronze arrowhead device in addition to the bronze service star.
Assuming he was an infantryman and/or served with an infantry unit in Alaska, he would have been assigned to any one of sixteen regiments - 4th, 17th, 32d, 37th, 53d, 58th, 87th, 138th, 153d, 159th, 184th, 197th, 198th, 201st, 297th or 364th - all of which served in Alaska at one time or another during the war. The 17th, 32d & 159th (later replaced by the 184th) were elements of 7th Infantry Division. All others were attached to Alaskan Defense Command or The Alaskan Department. The 364th (originally 367th) was a negro regiment; he'd have served with this unit only as a commissioned officer.
The only way to determine, with any certainty and/or accuracy, where he served and with what unit(s) is to access official records. I can do this but I have to charge a fee to recover my costs.