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We have conducted a new long-term multi-wavelength campaign on one of the most super- Eddington narrow-line Seyfert 1s (NLS1s) known, namely RX J0134.2-4258. In this first paper, we report deep simultaneous X-ray observations performed by XMM-Newton and NuSTAR in 2019-12-19, during which RX J0134.2-4258 was fortuitously at one of its lowest X-ray flux states. However, there is a clear rise above 4 keV which implies that the intrinsic source flux may be higher. The X-ray spectra observed between 1996 and 2019 show drastic variability, probably due to complex, variable absorption along the line of sight. Unusually, the soft X-ray excess appears extremely weak in all these spectra, even when the hard X-ray spectrum has a steep slope of $Γ\simeq 2.2$. We explore the spectral-timing properties of the new (low X-ray flux) and archival (high X-ray flux) XMM-Newton data, fitting their time-average, rms and lag spectra simultaneously. The variability spectra indicate the presence of a very weak soft X-ray Comptonisation component, whose shape is similar to the soft excess in normal super-Eddington NLS1s, but with flux relative to the power law which is lower by more than one order of magnitude. Above 4 keV the low-flux data are dominated by a different component, which lags with respect to the lower energy emission. This is consistent with an origin of reflection or partial covering absorption from low ionization material located within 100 Rg. We interpret this as a further indication of the presence of a clumpy disc wind.