What is the birth star of Goddess Radharani? There seems to be very few references of young Rukmini Devi. Was she at Vrindavan as Radharani during that time?
There is no basis for Rukmini and Radha being one.
Rukmini is daughter of Bhismaka, a king of Vidarbha.
Radha is daughter of Vrishabanu in Vrindavan.
Rukmini didn't see Krishna before Krishna kidnapped her to marry. It appears that she developed desire to have Krishna as husband only by hearing about him. Hence, she can't be Radha who saw Krishna in Vrindavan.
SB 10.52.23: Hearing of the beauty, prowess, transcendental character and opulence of Mukunda from visitors to the palace who sang His praises, Rukmiṇī decided that He would be the perfect husband for her.
SB 10.52.37: Śrī Rukmiṇī said [in her letter, as read by the brāhmaṇa]: O beauty of the worlds, having heard of Your qualities, which enter the ears of those who hear and remove their bodily distress, and having also heard of Your beauty, which fulfills all the visual desires of those who see, I have fixed my shameless mind upon You, O Kṛṣṇa.