Short-term Lamarckism in asexual single cell organisms
This phenomenon is well known and can be observed in several species. In fact, if you look at the time it takes for E. coli to change its transcription program in order to react to the environment (signal->transcription->translation), you will find it can be longer than its ~20 minute doubling time. You can indeed think of it as a form of Lamarckism. However, it is part of a field that has been booming in the last ~10 years, known as epigenetics.

Epigenetics is a field of research that deals with hereditary information that is passed by non-genetic mechanisms, i.e. not in the DNA sequence. This includes several mechanisms including passing of proteins/RNA, chemical modifications of DNA (methylation) and chromatin (histone modification). It turns out that epigenetic effects can be observed in virtually all species (including human) and affected phenotypes can be significant (leading to disease, for example).

This is a very interesting field of research and is very wide, so I would recommend reading a bit. Specifically regarding protein/RNA passing, I believe this is somewhat less deeply studied, perhaps due to the fact that the effect decreases exponentially (due to dilution), can be limited by protein/RNA stability and some measurement difficulties. However, I think I recall mathematical treatment of this in Prof. Uri Alon's book. You can also catch his systems biology course online on youtube, which I think touches on some of this material.