GRANDProto300 is one of the prototype experiments of the Giant Radio Array for Neutrino Detection. It will feature about 300 radio antenna detectors in Xiaodushang in Dunhuang, China, covering a total geometrical area of about 200 km$^2$. A main scientific goal of GRANDProto300 is the study of cosmic rays in the transition region (10$^{17}$ eV < 𝐸 < 10$^{18.5}$ eV). Our study calculates the exposure of GRANDProto300 to cosmic rays and estimates the number of cosmic-ray events to be detected during a fixed observation period. The trigger efficiency reaches 50, 80, and 90% at 10$^{17.5}$, 10$^{17.9}$, and 10$^{18.3}$ eV, respectively. The exposure of GRANDProto300 is 50 km$^2$ day sr at around 10$^{17.5}$ eV, and the expected number of observed cosmic rays with energies above 10$^{17}$ eV and zenith angles above 65$^{\circ}$ is about 130 events per day. GRANDProto300 will be able to measure the cosmic-ray energy spectrum in 10$^{17.2}$ eV $<E<$ 10^{19.5} eV through one-year observation, with a statistical precision about five times better than the previous spectral measurement by a mono-fluorescence detector of the Telescope Array Low-Energy Extension. The statistical uncertainty in the measurement of the mean depth of the air-shower maximum $X_{\rm max}$ is about five times better than the previous measurements using radio detectors at 10$^{17.5}$ eV; systematic uncertainties should be a dominant contribution limiting our interpretation of the chemical composition of cosmic rays in the transition region.