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This paper presents a BeppoSAX observation of NGC 7582 made during 1998 November and an optical spectrum taken in 1998 October. The new X-ray data reveal a previously unknown hard X-ray component in NGC 7582, peaking close to 20 keV. Rapid variability is observed with correlated changes in the 5-10 and 13-60 keV bands, indicating that a single-continuum component, produced by the active nucleus, provides the dominant flux across both bands. Comparison between RXTE and BeppoSAX data reveals changes in the 2-10 keV flux on timescales of months. Changes in the nuclear X-ray flux appear unrelated to the gradual decline in optical flux noted since the high state in 1998 July. The 0.5-2 keV flux of NGC 7582 is not significantly variable within the BeppoSAX observation but has brightened by a factor of ∼2 since the ASCA observation of 1994. While there is some concern about contamination from spatially unresolved sources, the long-term variability in soft X-ray flux seems most likely associated with the nucleus or an event within the host galaxy of NGC 7582. The 2-100 keV spectrum is well fit by a power law of photon index Γ=1.95^+0.09_–0.18, steeper by ΔΓ≃0.40 than the index during the 1994 ASCA observation. The X-ray continuum is attenuated by a thick absorber of N_H∼1.6x10²⁴ cm^–2 covering ∼60⁺¹⁰_–14% of the nucleus plus a screen with N_H∼1.4x10²³ cm^–2 covering the entire nucleus. Comparison of the BeppoSAX and ASCA spectra shows an increase in the full screen by ΔN_H≃7x10²² cm^–2 since 1994, confirming the absorption variability found by Xue et al. The increase in soft X-ray flux between 1994 and 1998 is consistent with the appearance of holes in the full screen, allowing ≲1% of the nuclear flux to escape and producing some clear lines of sight to the broad-line region. The data are also consistent with the scenario suggested by Aretxaga et al. of the radiative onset of a Type IIn supernova causing the observed optical change in NGC 7582.