The study addresses the growth of the new wild-type strain Cupriavidus necator IBP/SFU-1 and the synthesis of poly(3-hydroxybutyrate-co-4-hydroxybutyrate) (P(3HB-co-4HB) on media containing fructose and three different precursors of 4HB (ε-caprolactone 1,4-butanediol and 1,6-hexanediol). It was found that ε-caprolactone is the best precursor for the synthesis of P(3HB-co-4HB) copolymers. By varying the concentration and number of doses of ε-caprolactone added into the bacterial culture, it was possible to find conditions that ensured the synthesis of P(3HB-co-4HB) copolymers with different contents of 4HB (from 3−5 to 22.4 mol.%). The physicochemical properties of the copolymers were investigated depending on the proportions of 4HB monomers. The effect of 4HB monomers was manifested in a certain decrease in the weight-average molecular weight (Mw) (272−353 kDa), number-average molecular weight (Mn) (47−67 kDa) of the samples, and an increase in polydispersity (5.09−6.71) compared with P(3HB). The crystallinity degree decreased with an increasing fraction of the 4HB units (from 72 to 59%, as the 4HB content increased from 0 to 22.4 mol.%). In addition, the increase in 4HB content affected the temperature parameters (melting point, glass transition temperature, crystallization temperature, and thermal degradation temperature).