With the aim of understanding massive stars and their feedback in the early epochs of our Universe, the ULLYSES and XShootU collaborations collected the biggest homogeneous dataset of high-quality hot star spectra at low metallicity. Within the rich ``zoo'' of massive star stellar types, B supergiants (BSGs) represent an important connection between the main sequence and more extreme evolutionary stages. Additionally, lying toward the cool end of the hot star regime, determining their wind properties is crucial to gauging our expectations on the evolution and feedback of massive stars as, for instance, they are implicated in the bi-stability jump phenomenon. Here we undertake a detailed analysis of a representative sample of 18 Small Magellanic Cloud (SMC) BSGs within the ULLYSES dataset. Our UV and optical analysis samples early- and late-type BSGs (from B0 to B8), covering the bi-stability jump region. Our aim is to evaluate their evolutionary status and verify what their wind properties say about the bi-stability jump at a low-metallicity environment. We used the stellar atmosphere code CMFGEN to model the UV and optical spectra of the sample BSGs as well as photometry in different bands. The optical range encodes photospheric properties, while the wind information resides mostly in the UV. Further, we compare our results with different evolutionary models, with previous determinations in the literature of OB stars, and with diverging mass-loss prescriptions at the bi-stability jump. Additionally, for the first time we provide BSG models in the SMG including X-rays. Our analysis yielded the following main results: (i) From a single-stellar evolution perspective, the evolutionary status of early BSGs appear less clear than late BSGs, which are agree reasonably well with H-shell burning models. (ii) Ultraviolet analysis shows evidence that the BSGs contain X-rays in their atmospheres, for which we provide constraints. In general, higher X-ray luminosity (close to the standard $ (L_ X /L) -7$) is favored for early BSGs, despite associated degeneracies. For later-type BSGs, lower values are preferred, $ (L_ X /L) -8.5$. (iii) The obtained mass-loss rates suggest neither a jump nor an unperturbed monotonic decrease with temperature. Instead, a rather constant trend appears to happen, which is at odds with the increase found for Galactic BSGs. (iv) The wind velocity behavior with temperature shows a sharp drop at sim 19 kK, very similar to the bi-stability jump observed for Galactic stars.