Is it necessary to insert the pull up resistors between the SCL / VCC and SDA / VCC pins?
That is necessary but it must exist exactly once on the bus. As most breakout boards already include pull-ups you usually have to rather desolder them than to add. Because you failed to provide links to the components you use, you on your own to decide what pull-ups are needed or must be removed.
I connect the SDA/SCL pin of the oled to the A4/A5 pin of arduino and the SDA/SCL pin of the DS3231 to the other pin on the arduino SDA/SCL. I explain in more detail on the attachment.