Growing evidence suggests that epigenetic mechanisms, such as DNA methylation and demethylation, and histone modifications, are involved in the development of alcohol and drug addiction. However, studies of alcohol use disorder (AUD) that are focused on epigenetic DNA modifications and gene expression changes remain conflicting. Our aim was to study the effect of repeated ethanol consumption on epigenetic regulatory enzymes such as DNA methyltransferase and demethylase enzymes and whether those changes affected dynorphin/kappa-opioid receptor system in the Nucleus Accumbens (NAc). Two groups of male alcohol-preferring Alko Alcohol (AA) rats, rats which are selectively bred for high voluntary alcohol consumption and one group of male Wistar rats were used. The first group of AA rats had access to alcohol (10% ethanol solution) for 90 min on Mondays, Wednesdays and Fridays over a period of 3 weeks to establish a stable baseline of ethanol intake (AA-ethanol). The second group of AA rats (AA-water) and the Wistar rats (Wistar-water) were provided with water. Using qPCR, we found that voluntary alcohol drinking increased Dnmt1, −3a, and −3b mRNA levels and did not affect Tet family transcripts in the AA-ethanol group when compared with AA- and Wistar-water rats. DNMT and TET enzymatic activity measurements showed similar results to qPCR, where DNMT activity was increased in AA-ethanol group compared with AA-water and Wistar-water groups, with no statistically significant difference between groups in TET enzyme activity. In line with previous data, we found an increased percentage of global DNA methylation and hydroxymethylation in the AA-ethanol group compared with control rats. Finally, we investigated changes of selected candidate genes from dynorphin/kappa-opioid receptor system (Pdyn, Kor) and Dnmt3a genes that might be important in AUD-related behaviour. Our gene expression and promoter methylation analysis revealed a significant increase in the mRNA levels of Pdyn, Kor, and Dnmt3a in the AA-ethanol group, however, these changes can only be partially associate with the aberrant DNA methylation in promoter areas of the selected candidate genes. Thus, our findings suggest that the aberrant DNA methylation is rather one of the several mechanisms involved in gene expression regulation in AA rat model.