We report mapping observations of the ^(12)CO J = 3 → 2, 4 → 3, 6 → 5, and 7 → 6 transitions and the Ci ^3P→^3P_0 (Ci) 492GHz transition toward the central 40" × 40" region of the Circinus galaxy, using the Atacama Pathfinder EXperiment (APEX) telescope. We also detected ^(13)COJ = 3 → 2 at the central position of Circinus. These observations are to date the highest CO transitions reported in Circinus. With large velocity gradient (LVG) modeling and likelihood analysis we try to obtain density, temperature, and column density of the molecular gas in three regions: the nuclear region (D < 18" ~ 360 pc), the entire central 45" (D < 45" ~ 900 pc) region, and the star-forming (S-F) ring (18" < D < 45"). In the nuclear region, we can fit the CO excitation with a single excitation component, yielding an average condition of n_H_2~10^(3.2) cm^(-3), T_(kin)~ 200 K, and dν/dr~3 km s^(-1) pc^(-1). In the entire 45" region, which covers both the nucleus and the S-F ring, two excitation components are needed with n_H_2~ 10^(4.2) cm^(-3) and 10^(3.0) cm^(-3), T_(kin)~ 60 K and 30 K, and M_H_2~2.3 × 10^7 M_⊙ and 6.6 × 10^7 M_⊙, respectively. The gas excitation in the S-F ring can also be fitted with two LVG components, after subtracting the CO fluxes in the 18" nuclear region. The S-F ring region contributes 80% of the molecular mass in the 45" region. For the entire 45" region, we find a standard conversion factor of N(H_2) /I_(CO 1 → 0) = 0.37 × 10^(20)cm^(-2)(K km s^(-1))^(-1), about 1/5 of the Galactic disk value. The luminosity ratios of Ci and ^(12)COJ = 3 → 2 (R_(CI/CO 3 → 2)) in Circinus basically follow a linear trend, similar to that obtained in high-redshift galaxies. The average R_(CI/CO J = 3 → 2) in Circinus is found to be ~0.2, lying at an intermediate value between non-AGN nuclear regions and high-redshift galaxies
