Leaving aside for the moment, the original poster's readiness for first solo, let's review the nosewheel system and geometry.
A castering nosewheel means you have no control over it
Well, not exactly. A castoring nosewheel
with no steering cannot be directly controlled. A steerable nosewheel which castors, can be steered by the pilot, but will tend to castor if not steered. This would be the typical Cessna nosewheel system.
Look at a Cessna nosewheel with the plane parked. It slants forward, this is important. knowing that the steering is the freedom to rotate on the axis of the oleo suspension, and the fork arms do not cant aft (like a shopping cart wheel) or forward (like a bicycle), they are straight. Thus, the point on the tire, which corresponds to the center of steering rotation, is ahead of the point there the tire is contacting the ground. Thus the contact point, being behind the steering axis, tends to drag the wheel to follow the direction of motion (castor). The steering arms, which connect the pedals to the nosewheel have springs, which will allow the wheel to point differently than the pedals direct, though the compression of the spring tends it back to steer as desired by the pilot. Nosewheel Pipers I can think of do not have such a spring.
There are true full castoring only nosewheel planes which have no steering whatever, Grumman AA-1 through AA-5 series for example. Some brakes required.
When you and your instructor get further along in your training, and with suitable briefing before hand, try the following in the 152: With 10 flap, and into a gentle headwind, (on a grass runway, if you have a choice), and taking great care to not bang the tail tiwdown ring on the ground; hold the controls full back as you apply power for takeoff, and continue to hold it, until the nosewheel comes off the ground. Once off, let off a bit of the full back, to just hold the plane in that attitude. This will have happened by about 20MPH. Maintain steering with rudder only, it works perfectly fine, and you'll see that the nosewheel steering really does very little during a good takeoff. Oh, and be the way, you're about to do an excellent soft field takeoff too! Just don't bang the tail please!