For example - we need the population by single year of age from 12 years to 40 years for cities such as New Orleans, Atlanta, Indianapolis, San Diego, Minneapolis, Chicago, for the years 1975 to 2005.
Although we would love to have population by single year of age e.g. 12 yo, 13 yo, we could use data by age group if that is all that is available e.g. 12-14 or 12 to 15, 16 to 20, etc. Would need to know the county boundaries of the city population to be sure that it aligns with FBI data boundaries.
Finding the population by age, in principle, is possible since the AGE variable identifies all years of age. A challenge for this endeavor will be identifying specific cities within the years you are interested. In IPUMS USA the CITY variable identifies cities, but due to confidentiality restrictions on public use data, not all of the cities you mentioned are identifiable in all census years. The METAREA variable may be an alternative, but is a slightly different concept than a city and is also not identifiable in all years. Finally, US Census data up until the year 2000 only surveyed the US population every ten years.
An alternative may be possible using IPUMS CPS data. This source of data has annual data going back to 1962, however older samples have less availability of variables. The METAREA variable identifies most metropolitan areas for most years since the mid-1980s. One limitation with IPUMS CPS data with your project will be sample size. The Bureau of Labor Statistics notes that sub-state level analysis is prone to inconsistent estimates. This will particularly be the case if you are cutting your sample into age groups within metropolitan areas.