The study of short-term creep is essential for understanding the concrete creep process and deformation under alternating stress. Researchers are concentrating on the nano- and micron-scale creep of cement pastes. In the latest RILEM creep database, short-term concrete creep data at hourly or minutely levels are still rare and scarce. In order to describe the short-term creep and creep-recovery behavior of concrete specimens more accurately, the short-term creep and creep-recovery experiments were carried out firstly. The load-holding time varied from 60 s to 1800 s. Secondly, the accuracy of current creep models (B4, B4s, MC2010, and ACI209) in predicting the short-term creep of concrete was compared. It was discovered that the B4, B4s, and MC2010 models all overestimate concrete’s short-term creep, and the ACI model does the opposite. Thirdly, the applicability of the fractional-order-derivative viscoelastic model (with a derivative order between 0 and 1) in the calculation of the short-term creep and creep recovery of concrete is investigated. The calculation results show that the fractional-order derivatives are more suitable for analyzing the static viscoelastic deformation of concrete while the classical viscoelastic model requires a large number of parameters. Therefore, a modified fractional-order viscoelastic model is proposed considering the residual deformation characteristics of concrete after unloading, and the values of the model parameters under different conditions are given with the experimental data.