I will attempt to answer your questions with as much detail as I can be sure of. I will also be using the time period of New Kingdom Egypt, as that has some of the most detailed data on the lives of the people of Ancient Egypt. It also falls within the late Bronze Age that you asked about.

We have some images of a nobleman named Nebamon bringing his daughter on a hunt (some scholars say this may be symbolic). Being that this represents the upper nobility of Egypt, it's safe to say that it wouldn't be far off from what the Pharoah himself would do.

Another great look into the life an Egyptian Princess comes from the Amarna period and involves King Akhenaton and his daughters. We have intimate family reliefs that show him, his wife Nefertiti, and their daughters on Chariot ride through the city.

In the book "Red Land, Black Land", Barbara Mertz says that children of the time would play with rudimentary toys. That might include spinning tops, miniature weapons, etc. One Mechanical toy of note was a cat that had jaws which could be moved up and down by string. Wealthy nobles often had garden pools that allowed for swimming on their private estates.

They could also be expected to witness courtly entertainment such as the dancers and musicians that would play before the Nobles of Egypt. One Vizier's wife actually is actually picture playing a harp herself in one of our reliefs! Thus, perhaps playing the instruments was an option as well.

Banquets and parties were a standard part of entertainment, and general merriment seems to be the way that the rich would like to have passed their time. Drunkeness wasn't encouraged among the people, but it was known to happen That is a picture from Aidan Dodson's "The Tomb in Ancient Egypt".

One of a group of songs that we know as "the Harpers' Songs" had lyrics such as:

Spend the day merrily; put unguent and fine oil to your nostrils, and lotus flowers on the body of your beloved. Place singing and music before your face. Spend the day merrily, and weary not therein; lo, none can take his goods with him

So it would seem that having fun was a big part of life for the Egyptian people, and nobles in particular.

Now, on to the somber responsibilities of life :D. What you refer to as a "Princess" would be called a "King's Daughter" in Egypt. They might also serve the role of "King's Great Wife" which could mean that they had the responsibility to act as a sort of First Lady in Egypt. Or, they might've literally been married to their fathers... The good news is that the former is more likely to be correct due to the absence of children between the father and daughter relationships on record. "The Complete Royal Families of Ancient Egypt" By Aidan Dodson/Dyan Hilton goes into detail on the differing titles in the family and who actually held them in history.

Another big responsibility held out for a Princess was the position of "God's Wife of Amen". This was an influential religious position that required a ritual of masturbation for the statue of Amen in Thebes. The statue did travel at times, and in those instances the God's Wife might've followed. Amen was an extremely influential God with a Priesthood that would later on command a large part of the Egyptian population. The lands ascribed to his Priesthood would've been extensive. For more information on the role of a King's Daughter and the God's Wife of Amen, I'd recommend "The Woman Who Would Be King" by Kara Cooney and the article "Property and the God's Wives of Amun" by Betsy Bryan.

When it comes to education, we can't say that all princesses were tutored equally. We do have 2 scribal palettes that belonged to Princesses of the Amarna period though, so some obviously did learn to read and write.

The Harem that the daily life would take place in would be a large palace that would be able to accommodate hundreds of women. The Mittani Princess that was sent to Amenhotep III was said to be accompanied by some 317 lades in waiting. These ladies would likely have used much of their day weaving the nice clothes that a Princess was expected to wear.

Oh and speaking of foreign princesses, the Princesses of Egypt would not be expected marry outside of the royal family. At least till Ramesses II, royal women were forbidden to marry commoners. And probably my favorite quote from ANY King EVER is Amenhotep III's reply to King Kadashman if Babylon for a marriage alliance with Egypt by means of an Egyptian Princess: "From time immemorial no daughter of a king of Egypt is given to anyone." (emphasis mine) Mind you, he already had a couple of Babylonian Princesses for himself!

If you'd like more information, see publications such as "The Amarna Letters" by W.L. Moran (1992), "1177 BC" by Eric Cline (2014), "The Pharoah: Life at court and on campaign" By Garry Shaw, and "Ancient Egypt: Anatomy of a Civilization" by Barry Kemp.

Hope this helps!

Follow up question: Aside from who was doing the courting, is there any major difference to how a Prince would be spending his time?