We have measured the hyperfine structure and isotope shifts of the 402.0- and 399.6-nm resonance lines in Th-229(+). These transitions could provide pathways towards the excitation of the Th-229 low-energy isomeric nuclear state. An unexpected negative isotope shift relative to Th-232(+) is observed for the 399.6-nm line, indicating a strong Coulomb coupling of the excited state to the nucleus. We have developed an all-order approach to the isotope shift calculations that is generally applicable to heavy atoms and ions with several valence electrons. The theoretical calculations provide an explanation for the negative isotope shift of the 399.6-nm transition and yield a corrected classification of the excited state. The calculated isotope shifts are in good agreement with experimental values.