When using an open-drain output to drive an NPN transistor, can the base resistor be replaced by a pull-up?
Maybe, but you need to ensure that the Vol from the open-drain output guaranteed to be sufficiently low to ensure that the transistor is fully off under worst-case conditions with the value of R1 that you need to properly drive the base (typically something like the relay coil current divided by 10 or 20).
A guarantee of 300mV is usually sufficient.
Otherwise, you can use a pullup resistor and a voltage divider to the base, or maybe use a MOSFET, which could allow a higher value of R1.
For example, here are the typical characteristics of the open-drain outputs on the STM8S103F3 microcontroller datasheet (that I happen to have open at the moment):

None of the guaranteed numbers are directly useful, but combining the two it looks like 3mA would be safe with a 5V supply, so a 30-60mA relay could be driven as suggested.