Queen Morning Star | Princess Day Dreamer  
I thought a name changed would be alright to fit their roles as rulers but they can still go by their original name(s).  

 
Morning Star/Twilight Sparkle:
 
After the young Alicorn was ready to become a ruler of the sun, she had to pass one more test. To embody the Sun.
 
Twilight had to be ready to lead the world of Equestria with light and hope. Since Twilight had the power of friendship and was the element of Friendship she was mostly there.
 
Twilight passed her test and was kinda emotional. Her time has come.
 
The power of the sun was passed from Celestia to Twilight.  

 
Day Dreamer/Sunset Shimmer:
 
After some time living in the human world, Sunset had a sense to return to Equestria. This mostly accrued in her dreams.
 
At first, it didn’t make sense but she knew she had to go. When she said goodbye to her friends and entered the portal back to the pony world, Luna was waiting. Sunset had a choice to become Luna’s successor or return back to the human world to continue living her normal life. When Sunset was questioning what she wanted to do, Luna explained that her duties would be protecting ponies while they sleep during the night and during their dreams.
 
Sunset explained that she wanted some time to think. She wasn’t sure that she’d be a good ruler. Luna agreed to give her time and settle back into the Pony life.
 
Sunset eventually agreed to become Luna’s successor.
 
After some years of following Luna. Sunset was ready to embody the Moon. Sunset completed the Test and the power of the moon was transfused over from Luna to Sunset.
 
She met Twilight/Morning Star in the castle sitting on her throne. Both were shocked to see each other and was expecting Luna/Celestia to be in their places. After they both realized that they were going to rule together, they hugged each other and talked about their experiences on being a successor.  

 
Notes:  
So, when I was redesigning Twilight I wanted her to be the new princess/queen of the sun rather than being a single ruler. So I chucked in Sunset for three reasons;