If Brandon Stark had lived and Robert’s rebellion never occurred, would Ned have been appointed as Robert’s castellan of Storm’s End?

No way!

First, if Rebellion had not happened, Robert would still be just Lord Paramount of the Stormlands: This makes him inhabit Storm’s End like all other lords. It is possible that the castellan could be Renly Baratheon to give him something to do. Stannis Baratheon would obviously occupy a greater position as General of the Storm Armies or Rhaegar's future Master of Ships, so that is out of the question.

Without Rebellion, Brandon and Rickard are alive

Without the rebellion, Brandon Stark and Rickard Stark are alive, which means Brandon marries Catelyn Tully, as he originally would, eventually inheriting Winterfell and the position of Warden of the North. However, Ned Stark's future is a little more uncertain than most second sons of the Great Houses. Usually, second sons do not inherit land and are household knights. It makes one wonder if the Starks must have minor estates that have ruled through second son and cadet branches for centuries but are never mentioned.

Example: Karlon Stark and Karhold

Around 700 BC, a younger son of Winterfell named Karlon Stark helped to suppress a rebellion as a reward, he received some land where he built his fortress, Karls Hold. Over time, Karls Hold became Karhold, and Karlons descendants became House Karstark.

Considering Ned's Prestige

However, considering that Ned is the son of and brother of the next Lord Paramount of the North and is a personal friend of the Lord of the Stormlands, he has some extra prestige. It’s probable that he gets some lands and a castle without marriage.

Marriage Proposals

Regarding marriage, since Lyanna Stark is alive and was never kidnapped by Rhaegar, the Stark-Baratheon alliance already exists through Lyanna and Robert, as well as the Stark-Tully alliance through Brandon and Catelyn. It is likely that Ned would marry some damsel of the Vale to solidify the three kingdoms, perhaps an Arryn or Royce. However, having his marriage proposal refused by Aerys Targaryen, he could wed Cersei Lannister to solidify a future alliance with the North and its two allied kingdoms. Tywin knows the North is no joke; he could still have married Cersei to Oberyn Martell, but due to the age difference, Ned would be the better candidate, especially since Brandon, Rhaegar, and Robert are already on the throne.

Alternate Fanfic Endings

But to please fans of fanfics, it is possible that Ned and Ashara Dayne get married too. By all indications, the two fell in love at Harrenhal. Making an alliance with Dorne and uniting three Great Kingdoms in their family would not go down bad for Rickard. Ned and Ashara would possibly live in Starfall or High Hermitage, because she is a Lady. With Rhaegar on the throne in the future being a good king, they would spend most of their time at court, since Ned would have a lot of potential to be a Master of Laws or Coin, and Ashara would be a companion and friend of Elia Martell since her brother Arthur Martell and Rhaegar were also best friends.

Hope this helps!