"I remember this city [the] best" is the better sentence. There are any ways you can say this. "I remember it well, more than any other city I have visited" "This city brings back so many happy memories for me" "I remember this city the most" "I vividly remember this city more than any other" And lots more, hopefully other people will give their preferred suggestions and then you can choose, whichever you most prefer.

Russian speakers often put adverbs in the wrong place in English. Adverbs usually go at the end (after the verb, after any direct and indirect objects). Adverbs of frequency can go before the verb. For emphasis or literary effect, other positions are possible. I remember this city best. [standard position] John walked to work quickly. [standard position] John usually walked to work. [adverb of frequency] Usually, John walked to work, but not that day. [alternate position] A PDF of English Grammar in Use 4th edition by Murphy is available by a google search. See units 109 and 110.

Unlike some other languages, it is not uncommon in English to separate verb (remember) from adverb (best) in a sentence. So, the most natural-sounding option is, "I remember this city best." Please note that your two example sentences are in different tenses. The first one, with 'remembered', is past tense. The other one, with 'remember', is present tense. However, this difference does not change the best option. You could say, "I remember best this city." This construction keeps the verb + adverb together, and sounds more natural to me than the example you gave. However "I remember this city best" still sounds more natural.