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PKS 1413+135, a compact symmetric object (CSO) with a two-side pc-scale structure in its miniature radio morphology, is spatially associated with 4FGL J1416.1+1320 and recently detected with MAGIC telescopes. We comprehensively analyze its X-ray and GeV gamma-ray observation data for revealing its high energy radiation physics. It is found that the source is in a low-flux stage before MJD 58500 and experiences violent outbursts after MJD 58500 in the GeV band. The flux at 10 GeV varies ~3 orders of magnitude, and the GeV-flux variation is accompanied by the clear spectral variation, which is characterized as a soft log-parabola spectrum in the low-flux state and a hard power-law spectrum in the bright flares. The variability amplitude of X-rays is lower than that of gamma-rays, and no correlation of variability between gamma-rays and X-rays is observed. Fitting the broadband spectral energy distribution during a GeV outburst with a multi-zone leptonic model, we show that the GeV gamma-rays are attributed to the external Compton process while the X-rays are hybrid of several components. The predicted TeV gamma-ray flux during the GeV outburst is consistent with the detection of MAGIC telescopes. These results, together with its CSO radio morphology, imply that PKS 1413+135 has the episodic nuclear jet activities. The weak gamma-ray emission before MJD 58500 may be from its sub-pc-/pc-scale jet component powered by previous activities, and the violent outbursts with short-timescale variability after MJD 58500 could be attributed to the recently restarted jet activity.