Welcome to EP, Brian.
1) You will need a step-up converter or boost converter like
e.g. this one.
2) 35V*150mA=5.25W. Taking into account 90% effficiency of the converter (a conservative number which tells you that 90% of the energy input are available at the output) means your input power is 5.8W. A 10W solar panel is sufficient.
Note that the 10W for the solar panel wil only be achieved under optimum conditions. It will deliver considerably less on a cloudy day.
To get the maximum energy from your panel you can use an
MPPT controller, but that's probably overkill in your application.