He competed in the 1926 Wimbledon Championships and reached the fourth round in the singles event in which he lost to Jean Borotra. In the doubles and mixed doubles event he was eliminated in the second round. His second and final participation was at the 1929 Wimbledon Championships in which he reached the second round in all three events. He was a 28-times national champion of Denmark, five of which were consecutive singles victories. He played for the Davis Cup 74 times.
Ulrich got involved in the advertising industry when he established his own company Einer Ulrich Advertising in 1941 followed by Ulrich and Parrilds Advertising, which he sold to American James Walter Thompson where he remained as a co-director of the Danish department until his death. In the meantime he kept in touch with tennis as a chairman of the Hellerup Idræts Klub from 1938 and secretary of the Danish Lawn Tennis Association from 1929 and its president from 1964. He was also the editor-in-chief for the magazine Tennis. For a short time he was appointed the Denmark Davis Cup team captain. In 1967 Einer Ulrich foundation was formed, a project which goal was to help young tennis players. He married three times and had four children: a daughter named Kirsten from his first marriage with Karen Rigmor Larsen, a son named Arne Erlandsen and two sons with his second wife Ulla Meyer, also a tennis player and Danish champion. The sons became tennis players as well and were Davis Cup representatives, Jørgen Ulrich and Torben Ulrich. He married the third time to Rigmor Alvilda Landgreen. Einer Ulrich is also the grandfather of Metallica drummer Lars Ulrich, Torben's son.
After the Nazi occupation of Denmark in 1940 the family decided to stay in the country despite the Jewish origin of Ulrich's then-wife Ulla. The turning point was the year 1943 when they were informed of the Nazis' intent to "purge" the Danish protectorate as well. This rumor was strengthened by the deportation of the Danish police in 1943. In October the same year to prevent being arrested Einer used his ties to the Swedish KingKing Gustav V, a recurring sparring and doubles tennis partner of his, to send his wife and two sons to Sweden in secret. His family along with a group of other Jewish refugees were transported on a fishing boat by human traffickers across the Øresund strait when they were caught on the sea by the Germans. Shots were fired, the passengers jumped into the water and scattered. They were pulled aboard by the Nazis and taken into custody in Elsinore and then to a local camp. Einer made his way there and cleared up the situation with the German authorities and convince them to free his family. About six weeks later they decided to give the escape a second chance and this time they made it to Sweden with the help of probably-bribed customs officers. Einer joined them some time later with the help of another Davis Cup tennis player, Marcus Wallenberg Jr. of Sweden. After the war they moved back to Denmark.
