Dates between the years AD 201 and AD 299 in both the Julian and the proleptic Gregorian calendars are the same for the same day. Before 201 Julian dates are a little "later" than the proleptic Gregorian dates for the same day; for a random example 14 August 60 AD Julian is the same as 12 August 60 proleptic Gregorian. After 299 Julian dates are "earlier" than the proleptic Gregorian dates for the same day; for a random example, 14 August 333 AD Julian is the same day as 15 August 333 proleptic Gregorian.
To your specific interest in days near the start of the common era, The Julian January 3 1CE corresponds to the Gregorian January 1 1CE, according to this calculator.
The issue solved by the Gregorian calendar was that the Julian year was a hair longer than the tropical year (aka solar year - the time it takes for the sun to return to the same position, such as vernal equinox to vernal equinox). Over time, the errors had accumulated such that the Julian calendar date where any given yearly celestial event occurred had drifted by 10 days. In other words, even though the events were supposed to be yearly, the events were occuring at earlier dates every year. This was causing trouble for the Church which tied Easter to the vernal equinox but which had also pegged the equinox to Mar 21. During the era when the Gregorian calendar was developed, the equinox was instead occurring for the Church around Mar 11.
By having the Gregorian calendar start at the same time-point as the Julian calendar, but with a leap year algorithm fine tuned to be more in line with the celestial bodies, the Gregorian calendar moved the equinox back to Mar 21 (an ecclesiastically set date from back in the 3rd century, before the drift had become noticeable).
Different algorithms were suggested. One suggestion was to remove the leap years for the next 40 years. This would have caused no calendar date jump (no "days lost"), but would have lead to 40 years of special cases written into the algorithm forever. Instead, they chose to rip the bandaid off quickly. They decided to instead declare that they would switch from the Julian calendar to the Gregorian calendar (which both start on the same day but have different leap year algorithms) cold turkey:
When the new calendar was put in use, the error accumulated in the 13 centuries since the Council of Nicaea was corrected by a deletion of 10 days. The Julian calendar day Thursday, 4 October 1582 was followed by the first day of the Gregorian calendar, Friday, 15 October 1582 (the cycle of weekdays was not affected).
So, if you are using the Gregorian calendar, dates before 1582 will continue to follow the same pattern as the rest of the Gregorian calendar. If you think about it as though you are going back in time, the leap years will slowly cause the two calendars to sync until they are perfectly synced between 201 AD to 299 AD.