Does it absolutely have to be in the city? The Leipzig surrounding area is very well connected, and usually you can still get plots from the municipalities without developers at moderate prices. Otherwise, I can gladly recommend someone with expertise who can take a look at the construction specification and the contract.
It's always such a pro & con decision when you consider where to build, inside or outside the city. For us, the advantages of the city were clearly more significant, and in Leipzig you can still afford that with 500-600k. Besides, hopefully it's a kind of investment :)
And the plots outside the city don't cost much less if they are well connected to the city by S-Bahn. Only the infrastructure is usually significantly worse then, and the possibility to cycle to the city center (to work) no longer exists (I see this with a distance over 10km).
Any real estate agent would also confirm that a house in the city costs significantly more than the same house out in the sticks.