Yes. In general, the obligation to implement human rights does not depend on the wealth of a nation.
Especially the obligations to respect and protect human rights often do not require the commitment of large financial resources. This differs with respect to the obligation to fulfil human rights. This is why the International Covenant on Economic, Social and Cultural Rights stipulates that some economic social and cultural rights may be achieved progressively, depending on the existing resources. (Article 2, paragraph 1)
However, even then states have to act upon some obligations immediately. These include:
Most states will have to prioritise on how to implement human rights. Priorities have to be justified and backed up from a human rights prespective. (see question: “May a state prioritise a specific sector of development cooperation despite the fact that human rights are indivisible and interrelated?”)
Deutsche Geschellschaft für Internationale Zusammenarbeit (GIZ) (2013): Promising Practices - On a human rights-based approach in German development cooperation. GIZ, 26 p.
Deutsche Geschellschaft für Internationale Zusammenarbeit (GIZ) (2011): National Human Rights Institutions. GIZ, 10 p.