Just starting out with power, I expect that riding a bunch of miles to collect data points on your critical power curve will be more useful than estimating your FTP accurately. If you really want to know, then the most time efficient way I've found is to perform an indoor ramp test.
Also Strava isn't the only solution to tracking performance, far from it. I import my Strava activities into Golden Cheetah and get PMC and critical power estimates for free. If you use a Garmin, Connect also provides a power curve, as well as FTP and VO2Max estimates for free, though I suspect FTP might be based on a 20-min best power average.