I don't think so, microsoft's logic would probably be that giving Tomb Raider DE for free will give customers a tomb raider fix and they would be more likely to wait longer to purchase the sequel. The games are way too similar in terms of gameplay and graphics, if I got one for free I wouldn't be likely to spend $60 on a game that is almost identical. Imagine if you received Assassins Creed IV Black flag for $0, would you be likely to purchase Assassins Creed Rogue soon after for $60? Or would you rather instead spend that money on a game offering an entirely different experience. If microsoft wanted to hype people for the new tomb raider with the old tomb raider they should have given it away for free a long time ago.
Well, I don't think it will be out the month before but rather a few months in advance. I also don't agree with the 'more likely to wait longer to purchase the sequel', well, maybe if TRDE is given away free the same month it comes out but I don't expect that to happen. Majority of sales come from the first month, people want to play games when they're hot and I don't think making TRDE free several months before hand will change that. Also something to remember is that TR sold a ton more on PS compared to xbox, so MS may want to give the first game away free and at least make the exclusive deal worthwhile, an easy way to do that would be to get people hooked on the TR franchise is by making it free.
MS seems to like doing things like that, for example The Witcher 2 free last month before TW3(not exclusive but they have a marketing deal), Halo reach free in september before MCC, DR2 near the start of GWG before DR3. May just be a coincidence but it seems to be a trend for a few games so far.