Is it correct to say that a high variance feature will be important to my model?
High feature variance does not imply any sort of relationship to the target variable you're interested in modeling. Suppose you are looking at a population and have two variables for each person, their height in meters and annual income in dollars. The variance of height will be a small number (heights don't vary by much more than 1 meter), but the variance of income will be a large number (incomes can vary by thousands). If you want to predict the person's arm span, their height will be a much better predictor than their income, despite the fact that it has a numerically smaller variance.
A zero variance feature does not have any predictive power, but even numerically small fluctuations can be very important depending on your goal.