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It is hit-or-miss at first.  Once pointing is reliable, the only other variable is focus, and it can be surprisingly finicky.  The reason is that stars during the day are low-contrast objects, and if the focus is slightly wrong, you're smearing the light over a large blur circle.  At some point not too far from perfect focus, the object is lost in the bright-sky background.  In particular, focusing on some object 1 km away, then trying for stars, is not good enough.  Focusing on the Sun or Moon, or applying some known focus offset from some near object, is the ticket.

Pointing should normally be pretty good, even at 30x, because you have the measuring circles to help, and the instrument is leveled very accurately as part of setup (normally within 10 arcsec, better with reasonable care).  Acquire the instrument's azimuth offset using a limb of the Sun (I use eyepiece projection onto a white card), then use an ephemeris app to get topocentric coordinates of your star at a given time.  A planetarium app like Stellarium can help with mission planning.  Iron sights are of limited use because the object is only rarely visible to the naked eye (pretty much Venus only), but, as you say, they're fine at night.

I don't have a huge amount of experience with theodolite astro sightings, but for what it's worth, in broad daylight, anything brighter than mag 0 I find to be quite easy.  Polaris (mag 2), on the other hand, is more difficult, and I've managed it only when the sun is near the horizon.  This is with a 40 mm 28x telescope (Wild T2).  The great attraction of Polaris is that it moves very slowly, so it's easy to sneak up on it with the slow-motion controls, and the accuracy of timestamping is much eased---for run-of-the-mill stars or the Sun, you need 0.1 second or better, but with Polaris 1 second is fine.

A useful guide to theodolite astro observations is this pamphlet from Sokkia (it's a little dated):

http://www.rollanet.org/~eksi/2008EPHEM.pdf

 

Please be careful with eye safety.  Polarizers might not be opaque enough in the infrared, and the prospect of looking directly at the Sun through a 40 mm telescope with, possibly, only a few stops of attenuation in the IR is positively scary.  Sextant-shade manufacturers can be assumed to have done some due diligence here, but ad-hoc filters are taking a chance.  Try a full-aperture sun filter from Thousand Oaks or similar vendor; they're not too expensive.

Cheers,
Peter