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Start with a monotone nonincreasing function and sample it at finite set of points $x_0, ..., x_n$, $x_i<x_{i+1}$ so that $f(x_i)<f(x_{i+1})$. If you approximate $f$ with a linear spline then the resulting piecewise-linear approximation will certainly preserve monotonicity.

The question is: if you approximate $f$ with natural cubic spline will monotonicity be preserved as well?

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I assume you mean nondecreasing, i.e., monotonically increasing. If so, the answer is "no." For example, interpolate (0,0), (1,10) and (2,11) with a natural cubic spline:

enter image description here

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Not only is the answer "no", but for any number $N$ you can construct a monotone function and sample it such that the natural spline approximation will have $N$ extremum points.

See the figure (and its enlargement) below for an example. A full explanation of this function can be found in this answer.

enter image description here enter image description here

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