"to put up" something -- to build, to hang, to construct. Moomintroll was just hanging a swing when Sniff got home. Повесить чего то. Your understanding is very correct. In this sort of context the verb "to put up" means to hang or construct a thing vertically,
"to put up" with something -- to suffer through a challenge, to patiently react to your little brother, to deal with a minor series of problems. Терпеть / Потерпеть чего то