Copper sheath is the first choice for manufacturing high-Tc superconducting wires and tapes because of its high electrical and thermal conductivities, low-cost and good mechanical properties. However, Cu can easily react with superconducting cores, such as BSCCO, MgB2 and pnictides, and therefore drastically decrease the transport Jc. Here, we report the fabrication of Cu-sheathed Sr1−xKxFe2As2 tapes with superior Jc performance using a simple hot pressing method that is capable of eliminating the lengthy high-temperature sintering. We obtained high-quality Sr1−xKxFe2As2 tapes with processing at 800 oC for 30 minutes and measured high Tc and sharp transition. By this rapid fabrication, Cu sheath does not give rise to apparent reaction layer, and only slightly diffuses into Sr-122 core. As a consequence, we achieved high transport Jc of 3.1 × 104 A/cm2 in 10 T and 2.7 × 104 A/cm2 in 14 T at 4.2 K. The in-field Jc performance is by far the highest reported for Cu-sheathed high-Tc conductors. More importantly, Cu-sheathed Sr-122 tapes also showed a high Je value of 1.0 × 104 A/cm2 in 10 T at 4.2 K, which has reached the widely accepted practical level for applications. These results demonstrate that Cu is a very promising sheath for the practical application of pnictide conductors.