Not really. Still millions or billions of years. Unless they are very close, relatively.
You are missing the point. Are they sending a drone to every system? Why ours? They wouldn't be able to observe us with enough resolution to know we are here. The laws of physics don't change for them.
Actually, our science already is able to get the basics on nearby star system planets to determine if they orbit in the "Goldilocks zone" and have sufficient percentage of water vapor, oxygen, and carbon dioxide to support carbon based life as we know it here on Earth. Also, to detect other trace elements/compounds that would suggest possible industrial based activities. Of course the time lag of speed of light means we are seeing a relative past, based upon the light year distance, but even that could be useful information.
Even if one sends a vehicle that only reaches escape velocity from Sol/Solar System and then 'coasts' to some of the nearest stars, say within 4.5 light years (Alpha Centauri) or within a twenty light year radius (about 70 such stars), we are likely looking at decades to centuries depending upon final velocity(speed).
Here's a chart lifted from a book by Robert A. Heinlein; "Expanded Universe" (pages 447 -450)where in one of the chapters/articles he discusses travel times within our solar system to a couple of planets. The assumptions here are that the vehicles employ constant boast (steady impulse) acceleration which means increasing velocity/speed. This applies about halfway to destination at which point the vehicle flips 180 degrees and the thrust is now used to brake/slow down the approach. The times shown are round trip, there and back. We'll cover the details of the type/construction for such vehicles/vessels in another post. Also I may provide the math used in another post. Note the article these came from was written about 1979 ~ @ 45 years ago, in response to an earlier one from 1950.
ROUND TRIP BOOST COMPARISON OF ELAPSED TIME
Earth - Mars - Earth Velocity Earth - Pluto - Earth
4.59 days @ 1 gee 4.59 weeks
14.5 days @ 1/10 gee 14.5 weeks
45.9 days @ 1/100 gee 45.9 weeks
145 days @ 1/1000 gee 145 weeks
While Pluto isn't quite a light year out from the Sun, one could extrapolate these numbers to a 4.5 light year destination and see that even at 1/1000 gee acceleration=constant boost(impulse) it won't take "millions of years", let alone "billions".