Does my alphabet/neography have to have both upper and lower case?

Not at all! The majority of the world's writing systems don't have case distinctions; in fact, none of them did until fairly recently.

Originally, the Latin alphabet had no letter casing. The ancient Romans had one type of letter for formal inscriptions:

And another for handwriting:

And they usually didn't mix the two. Eventually, Roman cursive was replaced by other styles of handwriting, but those styles still didn't have case distinctions.

Eventually, though, a convention arose in illuminated manuscripts. When a wealthy person commissioned an expensive book, they would pay extra to have the first letters of sections and of important words decorated. (Decorating every letter would have been way too much—this was an era when a monk would produce about one page of text per day.)

And during the Renaissance, Humanist scholars codified this convention, basically by accident. They wanted to imitate the way the ancient Romans wrote, so they combined the ancient Roman formal inscription style for first letters with Carolingian miniscule (from the time of Charlemagne, not Caesar, but it looked so nice that it must have been Roman!) for all the rest of the letters.

The printing press caught on shortly after, so printers would make two cases (literal cases, as in wooden boxes) of type: the upper case (the case on top) for these first letters, and the lower case (the case on the bottom) for all the rest. And once it was standard in print, it stuck around forever. Greek and Cyrillic developed casing under influence from the Latin alphabet, which is why they do it in basically the same way. (And Cyrillic barely does; you just make the initial letters larger without actually changing their form.)

But as you can see, this is basically just an accident of history. Apart from Latin, Greek, and Cyrillic, most writing systems don't have any sort of casing. It's perfectly fine for a constructed one to do the same.