Exponential Fit Between 2 Lists

I guess the problem is that you do not provide an initial guess for the parameters, so as per the manual, curve_fit uses [1, 1] as a guess. The optimization might then get stuck at some local minimum. One other thing you should do is to change your xdata and ydata lists to numpy arrays, as shown by this answer:

import numpy as np
from scipy.optimize import curve_fit
exp_constants = np.array([62.5, 87.5, 112.5, 137.5, 162.5, 187.5, 212.5,
                          237.5, 262.5, 287.5])
means = np.array([211.94, 139.30, 80.09, 48.29, 26.94, 12.12, 3.99,
                  1.02, 0.09, 0.02])
def func(x1, a, b):
  return a * np.exp(b * x1)
guess = [100, -0.1]
popt, pcov = curve_fit(func, exp_constants, means, p0 = guess)

The exact value of the guess is not important, but you should probably have at least the order of magnitude and the signs right, so that the optimization can converge to the optimal value. I just used some random numbers close to the 'correct answer' you mentioned. When you don't know what to guess, you can do a polyfit(xdata, log(ydata), 1) and some basic math to get an initial value, as shown by this answer to the question you linked.

Quick plot:

x = np.linspace(exp_constants[0], exp_constants[-1], 1000)
plt.plot(exp_constants, means, 'ko', x, popt[0]*np.exp(popt[1]*x), 'r')
plt.show()

Result: