Coronavirus disease 2019 (COVID-19) is a new global pandemic with high morbidity and mortality caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). N-Hydroxycytidine derivatives show promise for combating COVID-19 and other viral diseases; in particular, molnupiravir has recently been approved for emergency prophylaxis in the early stages after infection with SARS-CoV-2. In the present work, a scheme for the synthesis of 5-halo-2ʹ-azido-substituted cytidine and N-hydroxycytidine derivatives was proposed. The synthesized compounds were tested against a panel of six RNA viruses, including SARS-CoV-2, enteroviruses, CHIKV, and HIV-1. A number of compounds showed ability to inhibit the reproduction of SARS-CoV-2 and CHIKV viruses in the micromolar range without noticeable cytotoxicity. The structures of the leader compounds can be used as a starting point for further design of antiviral agents.