A 1 liter starter is not big enough for a Doppelbock. You need more like a gallon starter. It will still make beer, but the lag time may be significantly long and you will likely produce some noticeable off-flavors from all the yeast growth. I usually make my lager starters up to 2 weeks in advance. Between stepping them up and crash-cooling them, they can take that long.
As for the starter temperature, there are differing schools of thought on this. Some say you're only making yeast, so room temp is fine. Others say fermentation temp is best because the lager yeast can get used to fermenting warm (and produce esters, etc.) and then continue to produce some of these esters in the main ferment. I like to keep the starter ferment at or under 60°F for lagers. One general rule is that if you do ferment the starter at room temp, you should crash-cool and decant off the starter beer before pitching... otherwise the esters produced in the starter will affect the taste of your beer. Crash-cooling the starter takes longer for lager yeasts because they remain active at colder temps. 2 days of crash-cooling is usually enough.