NGC4589, a bright E2 merger-remnant galaxy, hosts the peculiar fast and faint calcium-rich supernova SNIb SN2005cz. The progenitor of Ca-rich SNeIb has been controversial: it could be (1) a young, massive star with 6-12M{sun} in a binary system, or (2) an old, low-mass star in a binary system that was kicked out from the galaxy center. Moreover, previous distance estimates for this galaxy have shown a large spread, ranging from 20 to 60Mpc. Thus, using archival Hubble Space Telescope/Advanced Camera for Surveys (HST/ACS) F435W, F555W, and F814W images, we search for star clusters in NGC4589 in order to help resolve these issues. We find a small population of young star clusters with 25<V<=27(-7.1<M_V_<-5.1)mag and age <1Gyr in the central region at R<0.5' (<3.8kpc), thus supporting the massive-star progenitor scenario for SN2005cz. In addition to young star clusters, we also find a large population of old globular clusters. In contrast to previous results in the literature, we find that the color distribution of the globular clusters is clearly bimodal. The turnover (Vega) magnitude in the V-band luminosity functions of the blue (metal-poor) globular clusters is determined to be V_0_(max)=24.40{+/-}0.10mag. We derive the total number of globular clusters, N_GC_=640{+/-}50, and the specific frequency, S_N_=1.7{+/-}0.2. Adopting a calibration for the metal-poor globular clusters, M_V_(max) =-7.66{+/-}0.14mag, we derive a distance to this galaxy: (m-M)_0_=32.06{+/-}0.10(ran){+/-}0.15(sys) (d=25.8{+/-}2.2Mpc).