corolla in this budget will be available but the model will be quite low like 2000-2002 max. whereas honda city you can easily get 2004-2006 model. that car requires less maintenance if driven properly. in my opinion, try to find a good reliable mechanic or a workshop and then buy a normal condition car and modify/restore/maintain as per your own liking.
if you like get a honda civic EXI 1996-2000 model you may easily get it in about 6-7 lacs. similarly, you may get 2001-2004 model EXI in the range of 7.5-9 lac and custom build according to your own taste. this will last long maintenance-free for a long. regular maintenance like oil change and filter change would be required other than that might not be needed. fuel economy you can get easily within a city would be around 12 and long routes easily can get around15++. this will be more comfortable and hassle-free as compared to honda city.
if you want to buy a corolla then try to find an Indus corolla model would 1992 till 2000. for this car, the price varies a lot in the market. you may even see people asking 12 lac for it. but you may find a normal condition car in around 7-8 lac from a small city. spend around 2-3 lac for complete restoration and then that car would serve you easily for the next 4-5 years hassle-free.
The same goes for the next generation corolla like from 2001 onwards but that would be really hard to find even a normal condition one under 10 lac so that you have the remaining amount for up-gradation.
in case of any further help just inbox me.
thanks.