How can I make Siri's geo-fenced reminders wait until I actually arrive?
You don't have precise range control over geo fenced reminders with non iBeacon input. Both GPS and WiFi and cell tower location can trigger locations a good block or more away from a precise location.

Until you have iBeacon hardware or software set up, you can snooze that reminder, set a follow on time based reminder or change the original reminder to go off time rather than proximity. 

The best you could do is offset the home location to coordinates several hundred feet away from where you typically approach and keep the same default 100m/330ft radius that iOS lets you make larger, but not smaller on iOS 7.