Finally, PlayStation Store is slowly catching up to the features offered by Xbox Live Marketplace. With the new Download Queue option, you'll be able to remotely manage your PS3 and Vita download list for the next time you turn your system on.
As PSN Stores points out, this feature was enabled in PS3 firmware 4.40. However, the web store didn't support queuing until just now.
Your system won't turn on automatically and download your content. Instead, it will start downloading the content once you turn on your system. A message: "You have content items in PlayStation Store that are ready to download" will appear. It's not the most elegant solution, but at least it's something. (Auto-downloads may be possible for PlayStation Plus members that have automatic updates turned on. However, they'll still have to wait until their designated time for the downloads to start.)