Depending on the fuel, rockets can either be simple or complex, less or more efficient. But for a decent rocket that could get something to space, and indeed was used as the model for the Saturn V, let's look to the V2 rocket. That's second world war period (1944). So, look at around that time for rocket technology. You are not just looking at pure thrust, you also need to consider guidance systems. Hypothetically, you could generate the thrust, but if you don't have the technology for the guidance system and control systems, then you may as well just fire it and cover your eyes - hoping it won't hit you.
However, you want to also send a probe. Now, you have to ask yourself: "what kind of information do I need this probe to gather?" a 1kg probe is not large and is limited in the telemetry that it can carry, and thus would be limited in the science that it can perform. Particularly if you are looking at a very early development probe. But there's a lot to consider here.
A probe needs a power source, early batteries were large and heavy. These batteries needed to power not only the transmitter (to send whatever data you're gathering), the data gathering system (gets whatever data you are interested in), and the thermal regulation system (because space is a vacuum, it's actually pretty hard to cool down. Most thermal transfer on Earth is done via convection and conduction. Without air, you're stuck with radiation.)
So, already you're looking at something probably way bigger than 1kg. Of course, you can reduce this weight by ditching a science device and just having a transmitter that basically just beeps so that the ground station knows that it's actually up. You would never send up a probe without this, you may as well just launch an empty rocket.
Remember the more primitive the technology, the bigger and clunkier stuff is. Transistors on circuit boards were larger. Batteries were larger.
Have a look at Sputnik as an example. That big ol' sphere with sticks was a whopping 80kg (or something).
The Space Race caused massive leaps in technology. The huge number of people who worked for the space organizations were often breaking ground and setting new paradigms. So, if your civilisation suddenly decided to totally focus on getting something into space, maybe they would have that huge leap in technology - and all the useful payoffs for general society's technology, particularly once they start launching people.
I love speculative questions like this (: I'm an aerospace engineering master student, so that's where a lot of my info has come from.