We develop a small-signal measurement system on cryogen-free dilution refrigerator which is suitable for superconducting qubit studies. Cryogen-free refrigerators have several advantages such as less manpower for system operation and large sample space for experiment, but concern remains about whether the noise introduced by the coldhead can be made sufficiently low. In this work, we demonstrate some effective approaches of acoustic isolation to reduce the noise impact. The electronic circuit that includes the current, voltage, and microwave lines for qubit coherent state measurement is described. For the current and voltage lines designed to have a low pass of dc-100 kHz, we show that the measurements of Josephson junction's switching current distribution with a width down to 1 nA, and quantum coherent Rabi oscillation and Ramsey interference of the superconducting qubit can be successfully performed.