Which resistor for NPN transistor base?

If you want to conduct up to 150mA of current between node 1 and ground and only drop 1.0V (it's not a perfect switch) then you'll need to assume the current gain is typically 50. If you want a lower saturation voltage then the spec sheets says feed the base with 15mA i.e. the current gain has dropped to only 10 but saturation will only be 0.3V.

So assuming you are happy with 150mA while saturating the transistor to about 1V, you need to push \$\dfrac{150mA}{50} = 3mA\$ into the base.

The base voltage will need about 0.7V so the remainder (1.8V - 0.7V) needs to be across the resistor R1. Ohms law tells us that R = \$\dfrac{1.1V}{3mA} = 366.7\Omega\$.

So choose maybe a 360 ohm resistor.

If this isn't good enough for your needs look for an N channel MOSFET with low \$V_{GS(threshold)}\$ - something like 1V or less.

Current

There's a rule of thumb that you make the base resistor such that the base current is a good multiple (5, 10?) of the minimum base current needed to put the BJT into saturation - you can get this from the data sheets.

Voltage

You need to know \$V_{BE(sat)}\$ - but that's in the data sheets too. It's probably safe to assume 0.7 V for most small-signal NPN BJTs.

Resistance

Once you know the voltage across the resistor and the current through it, Ohms law will tell you the answer.