In recent years, dual repetition-rate mode-locked lasers with slightly different pulse repetition rates, as newly developed ultrafast lasers, have attracted great interest and shown their applications in ultrafast dual-comb spectroscopy, asynchronous optical sampling without mechanical movement, etc. The traditional dual-comb system composed of a pair of independent optical frequency combs with slightly detuned comb spacing is still considered expensive, complex and fragile. It is imperative to develop practical and compact dual-comb devices. Dual repetition-rate ultrafast lasers generating asynchronous ultrafast pulses directly from a single cavity can be a promising alternative to the current dual-laser-based comb source. A dual-comb setup based on single laser has the advantages of compact structure, low cost and intrinsic mutual coherence. This technique paves the way for developing the compact, robust and environmental-immune dual-comb systems. In this paper we develop an alternative dual repetition-rate mode-locked Yb:YAG ceramic laser that emits a pair of pulses with spatially separated beams from a single cavity by using a semiconductor saturable absorber mirror and a dual-path pump configuration. In our experiment, a high quality transparent Yb:YAG ceramic prepared by non-aqueous taper-casting method is selected as the gain medium, which is pumped by a 940 nm laser diode. A dual-path pump configuration consisting of a pair of polarization beam splitters and a pair of half-wave plates is designed, in which total pump power from a laser diode is divided equally for pumping the two separate laser beams. When the total absorbed pump power is 5.6 W, dual repetition-rate continuous mode-locked laser operation is achieved under the gain-loss balanced cavity condition. The pulse repetition rates of Pulse1 and Pulse2 are 448.918 MHz and 448.923 MHz, respectively. The difference between repetition rates is 5 kHz mainly caused by the different optical path lengths in the cavity. Under an absorbed pump power of 7 W, the maximum total output power extracted from this laser reaches 170 mW, i.e., 89 mW for Pulse1 and 81 mW for Pulse2. The two mode-locked pulses have nearly identical spectral shapes centered at 1029.6 nm and 1029.8 nm, respectively. The spectral bandwidths for Pulse1 and Pulse2 are 1 nm and 1.16 nm, respectively. The corresponding pulse durations are 2.8 ps and 2.6 ps for the Pulse1 and Pulse2 respectively. Our scheme integrates the advantages of self-starting operation, high repetition-rate, suppression of gain competition. These results indicate that dual-path pump configuration is feasible for dual-repetition-rate mode-locked lasers. These co-generated, dual repetition-rate pulses from one laser cavity possess similar laser characteristics and can be operated independently by dual-path pump configuration. This laser has potential advantages of compact, cost-effective and high-stability for single-cavity-based dual-comb applications in dual-comb spectroscopy, distance ranging, etc.