There are a few ways to interpret "the orbits are similar to the current Earth-Moon system". We can keep the Sun-Earth and Earth-Moon orbit distances the same, or we can keep the orbit periods the same.
The orbital period equation for a two body system is
$$R^3 = \mu (T / 2\pi)^2$$
where $R$ is the semi-major axis, $T$ is the period, and $\mu$ is the sum of the standard gravitational parameters of the two bodies. That equation can be derived from the vis viva equation. (Also see Specific orbital energy).
For a small body orbiting a much larger body, we usually ignore the mass of the small body, but that's not very accurate when the bodies have similar masses. To keep the calculations simple, I'll assume that all the orbits are circular (and coplanar), so the orbit radius equals the semi-major axis.
I decided to keep the current Sun-Earth distance and to use the current sidereal period of the Moon for the period of the replacement moon. So we have the Earth and planet orbiting each other in a circular orbit, and the barycentre of that Earth-planet system orbiting the Sun in a circular orbit. Because the Sun is so massive, the year length of the modified system is fairly close to the current year. The biggest difference is (of course) with Jupiter as the moon, which gives a sidereal year of ~365.082 days.
However, the Earth-planet orbit radius changes significantly, especially with the giant planets.
Here is a table of the resulting Earth-moon orbit radii and angular diameters, as seen from the Earth's equator. The radii are given in units where the current Earth-Moon radius is 1. The angular diameters are in arc-minutes. I used the planets' polar radii for the angular diameter calculations.
FWIW, the barycentres of these orbits are between the Earth and the other body, except for the Moon, Jupiter, and Saturn.
Here is the Python script which I used for those calculations. The gravitational parameters and polar radii came from Horizons body data. The script prints a few more values, including the year and synodic month lengths, and the distance of each body from the barycentre.
We can use those angular diameters to estimate the eclipse duration relative to eclipses in the current system. But we can't estimate the duration of totality without knowing the new Earth's day length. With the Earth orbiting a giant planet, it's likely that it would be tidally locked, so the day length would equal the month length.
The solar eclipses caused by Mercury and Mars would be similar to current solar eclipses, although some sunlight would be refracted by the thin atmosphere of Mars. The other planets have a lot of atmosphere, so they would refract a lot of light, which would create a similar effect on Earth to what we see on the Moon during a lunar eclipse in the current system.
The radius values for the giant planets are based on where the atmospheric pressure is equal to 1 bar, i.e., Earth's surface air pressure. There's still quite a lot of atmosphere at higher altitudes. And of course if the giant planets were only 1 AU from the Sun they would be significantly warmer, which would affect their atmospheres.